Talk:Descartes' theorem

2033:– the whole section you just removed is a special case of the generalization of Descartes theorem. The discussion was focused on the algebraic identity which is directly analogous to the "Descartes theorem" identity for the mutually-tangent-circle configuration. The purpose of including it was to (a) demonstrate how the generalization works in a specific case, and (b) show how the two "soddy circles" (related by the ± cases of the find-the-fourth-curvature version of Descartes' theorem) are strictly related to each-other as well as related to the other three circles, while (c) also showing how Descartes' theorem is related to nearby topics in inversive geometry. All of that is directly relevant to this article. I don't see how removing it benefits readers. We don't have a page limit here, but even if we did this article is not excessively long or sprawling. Many (most?) readers probably won't read all the way from beginning to end, but it's not like they're being subjected to an arbitrary snorefest mishmash here, and some readers will likely find the section relevant and interesting. – 2187: 138: 2070:. All the rest of the article is filler. Because the part I want is buried two levels down somewhere in the middle of the article, it always takes me much longer to find it than it would for a better-focused article. All the rest is literally wasting my time, repeatedly, every time I want to refer to the article. It's annoying. I don't want this article to be equally annoying by becoming buried in stuff you don't want to look up to the point where finding the parts you do want to look up becomes difficult. — 128: 107: 3491:– In the event that the three circles are all mutually tangent, then yes it is always non-negative. The case where they are externally tangent has all of the ks positive. The case where two circles are nested inside another the product of the ks for the two smaller circles must be at least as large as the sum of products of each with the k for the circle enclosing them, with equality when the two internal circles each have half the radius of the enclosing circle. Then 4210:

notes for different types of textual notes. Others use numbered or author/year parenthetical references referencing a bibliography list along with textual notes using some other format. Some use per-chapter endnotes while others put all the notes at the end of the book. Some bibliography lists are very compressed auth. (yr.) journ. abbrev., vol.: pg., with no paper title. Others include detailed annotations about each bibliography entry. Etc.

21: 1868:, and you can find them by applying a transformation that turns all three of your original circles into points, taking the two circles passing through those points with opposite directions, and invert the transformation to recover the two circles you are looking for. (If you want to find all 8 solutions to the un-oriented circle problem, you have 4 choices for the relative orientations of the 3 source circles.) 3807: 3797: 3784: 3771: 3758: 3748: 3730: 3716: 3702: 3688: 3662: 76: 4466: 267: 3825: 3675: 1853:

envelope of lines, and so on), so spherical Laguerre transformations can be represented, just like spherical Möbius transformations, as fractional linear transformations of complex numbers; but the Euclidean plane has circular angles and flat distances, so the dual Laguerre geometry switches these, and Laguerre transformations are modeled by fractional linear transformations of

216: 2815:(which by the way is one of the GA requirements): We should not make the article more technical than it needs to be. Signed areas are an unnecessary technicality. As for P_4 at infinity: let's not get into trying to set up equations of equal area of triangles that have a vertex at infinity. That way lies original research and dubious definitions and confusion. — 3229: 1849:, which preserve line–circle tangencies and preserve distances along lines between them but do not preserve points (for example two intersecting circles might map to two non-intersecting circles under a Laguerre transformation, but the pair of lines each tangent to both original circles will map to a pair of lines tangent to both resulting circles). 1588: 943:

yields one solution while using a − sign in both equations yields the other. In other cases, both of those are wrong; one solution is found using a + sign in one equation and a − sign in the other, while the other solution is found by swapping those signs. I haven't yet determined what makes the difference.

4194:

Sure, there are many ways to do this, and we do not have to observe the paper-based dichotomy of "explanatory footnote directly on the page, citations referring to bibliography at the end" (it would be nice to have to have support for this distinction for printed Knowledge articles, but I don't think

1169:

I agree with this. An article about a theorem should do a bit better job showing or at least explicitly linking to proofs. The history section mentions that "Multiple proofs of the theorem have been published" but does not directly cite any of those inline, and the few-words summaries are unhelpfully

459:

can transform tangent circles into parallel lines or vice versa. So with these conventions it is indeed possible for three lines to be mutually tangent: just make them all parallel. However in this case it would only be possible to make the fourth thing that is tangent to all of them be another line,

425:

In par 3 under Special Cases: "...as it is not possible for three lines and one circle to be mutually tangent." - my immediate reaction was 'incircle of a triangle?' Probably should be clarified that mutual tangency of three lines makes no sense. But then the previous case of two parallel lines being

4106:

As for expanding our quote of the poem later in the article: I'm not entirely persuaded that we have a valid fair-use case for doing so, and I think the quote is part of a summary of later material already: it is used to point towards the discussion of the poem in the history section, but it is also

4062:

for helping with the article! Formal recognition or brag sheets are indeed secondary to article improvement (getting a thorough GA review is more important to me than having another green plus to display on my user page). But it can help slightly with your standing inside Knowledge so it is not just

1789:

The general concept surely dates from the 19th century. Who came up with the version in the Knowledge article? I think they messed up with the name; it could be called the "inversive circle dot product" or "circle cosine" or something (Kocik uses the name "Pedoe product"), as even though distance is

4213:

If someone wanted Knowledge articles to render better for printing it would be possible to make with CSS, but a significant challenge to handle all of the range of possible edge cases. (It would be great to see a same-page footnote version, with the notes repeated any time they are referenced again

4209:

I'm not sure which "paper-based dichotomy" you mean. Books and research journals have an even wider range of practices than Knowledge articles. Many freely mix bibliographical information with textual notes, using either footnotes, sidenotes, or endnotes. Some use a combination of footnotes and end

4008:

can be interpreted based on the inverse (Euclidean) diameter under stereographic projection (conformal disk model). It would be nice if we had an article or section somewhere on Knowledge describing more clearly what curvature of a curve or generalized circle means in spherical/hyperbolic geometry,

2829:

To be clear: I am not suggesting we should try to get fancy about areas in the straight line case. It just falls between the other two (I guess I might also mention one additional case when the center is on the triangle and one of the triangles degenerates, with area 0). Maybe the clearest for this

1628:

of each pairs of circles in the resulting configuration (including of each circle with itself along the diagonal), and by changing the coefficients in the matrix we can enforce different conditions on the circles. (I got tripped up a bit because the particular "Descartes configuration" matrix above

4148:

Re potentially relegating renumbered footnotes and (consisting of both a citation and some text describing how the citation supports the cited claim) into a separate notes section and then splitting off the citations from these footnotes into footnotes-within-footnotes: see ongoing discussion at

2011:

because then readers can very easily see how this matrix equation is equivalent to "equation 1" from earlier in the article. It seems to me that taking that part out makes that section of the article slightly more compact but substantially harder for semi-technical readers to follow (say someone a

2133:

I would have no objection to similarly repurposing the removed subsection on Steiner chains as part of another article. I don't think it's bad content, in general. It just seemed overdetailed regarding stuff that is not the relation between curvatures of four tangent circles for inclusion in this

1174:

have you done any survey of the published proofs of Descartes' theorem? Can we make a section including a couple of them or at least providing a sketch? I feel like a section about proofs of this theorem (and maybe some more images for the history section) is the only thing really missing to make

1793:

Coxeter's version of "inversive distance" apparently only considers non-intersecting circles and takes the arccosh of the absolute value of the Knowledge article's concept, which is more like a distance, but has the problem that it erases the orientation of the circle from consideration and also

942:

I've been writing a program to generate Apollonian Gaskets, and I've come to a minor hiccup. It seems the ± sign in Equation 2 does not correspond to the ± sign in the unnumbered equation immediately after Equation 4 (Complex Descartes' Theorem). In some cases, using a + sign in both equations

4177:

Personally I freely add text to "ordinary" Knowledge footnotes any time it seems like the text would be helpful to readers but is distracting or out of scope for the main body text, on all sorts of articles, most of which have no separate "textual notes" section distinct from the "bibliographic

1852:

On the sphere, where both distances and angles are circular, there's a precise duality between the two (the dual of every point is an oriented great circle, and vice versa; every triangle has a dual "trilateral", every spherical conic as a locus of points has a dual spherical conic which is the

1813:

but figuring out what the scope and focus should be for an article and how to organize it is a bit daunting, since this is a topic which appears regularly here and there but is uncommon enough to have never quite crystallized into part of the canon or have particularly widely adopted or settled

1874:

As an example of a practical application, a "circular mesh" is a quadrilateral mesh where every face is a cyclic quadrilateral, a property preserved under Möbius transformations. A dual idea, the "edge offset mesh" has every edge emanating from a vertex lying on a common right circular cone, a

4129:

Personally I don't really see an issue with the current use of the poem in the lead. The line from the poem is mainly used as a convenient device for stating the theorem in English words instead of introducing a bunch of symbols in the lead section; the full text of the poem isn't really that

1090:

Answering myself: After re-reading, I realize that my paraphrasing is wrong too. My apologies. I think that the clarification needs to be made in the following sentence, though. Something along the lines of "By solving this equation, one can start with three given mutually tangent circles and

4026:

By the way, since I don't see it anywhere here: jacobolus made significant improvements to the article in the lead-up to its GA nomination and should get credit for that if this passes GA. I nominated this myself but am happy to see jacobolus's continued involvement in the GA review process.

454:

It's not really true that "mutual tangency of three lines makes no sense". In the geometry of circles and lines it makes sense to think of lines as a limiting case of circles with infinite radius, and to think of parallel lines as being a limiting case of tangent circles. For instance, a

4012:

I guess the image could perhaps seem a bit out of scope insofar as we don't give any examples of kissing (generalized) circles in the hyperbolic plane, which might make it seem like unmotivated detail in a section which is otherwise quite condensed. I still think it's helpful, but YMMV.

2058:

is another. Again, we have an article about it, where more detailed material can go. The fact that you can measure the inversive distances of arbitrary circles and put them into a matrix is not particularly informative. And again, it's not particularly on-topic here. In that sense, the

3956:

For the poem, Soddy died in 1956 so it will become free on 1 January 2027. Citing four lines instead of two would still be perfectly acceptable though. Gosset died in 1962 so it is 1 January 2033 for his poem. Awaiting your responses to my other points (none of them showstoppers).

4458:

hooks for this article – which itself has well-sourced and neutral text, and appears to be free of copyvio. It also has great images, although those are outside the scope of this review. As the article and hooks meet DYK criteria, this is good to go! Very nicely done as usual,

3354:

The poem doesn't appear in the body; generally I prefer not to have extra stuff in the lead. Maybe you can have a little more of the poem in the body, perhaps something explaining about positive and negative curvatures or how "a fifth sphere in the kissing shares" in the 3d

4144:

Re "perhaps use again as in 15b?": done (causing footnote renumbering). Re " could be used to clarify that Gosset was not the only one to generalise to higher dimensions": this was already done when the Gardner reference was added: "Gosset and several others extended the

3902:

Sources are fine with just tiny comments, no original research, no problems of copying or close paraphrasing detected. Will put on hold just so you get a notification that I'm done reviewing, not because I think it will take a long time to fix the little issues I found.

1395: 2506:

is outside all three angles of the triangle (in the interior of one of the vertical angles. The easiest way to deal with all of these is to allow a negative radius, but it might be unnecessarily confusing. The next easiest way is to note that the sides adjacent to

4217:(Sorry if this sounds argumentative or seems off topic. I guess my main point would be that I don't think GA criteria should push for change between otherwise acceptable citation formats unless there's some very obvious issue causing trouble for readers.) – 843:

centered on one of the points of tangency; the resulting configuration will consist of two parallel lines with a circle sandwiched between them, at which point it should be obvious that there are two choices for the fourth circle. Actually, according to

3977:, the reason I made an image like this is that it's not immediately obvious (at least to me) that the kissing "circles" in the hyperbolic plane can include not just circles but also horocycles, hypercycles, and geodesics, and that the resulting "bends" 2910:

A construction of what, from what givens? If you are just going to make me guess what you mean by repeating the same question in the same words that you already asked, I am not going to be able to give you a better answer than the one I already gave.

4241:'s "significant information should not appear in the lead if it is not covered in the remainder of the article" as the poem is covered, even if without quotes. That said, I do hope there will be more of the poem in the article once it becomes PD. — 2882:, which we describe early in the history of this section. It is the attempted reformulation of this problem using algebra rather than compass and straightedge that led Descartes to Descartes' theorem, as we also describe in the history section. — 1191:

algebraic and unsatisfying proof that I encountered. They all seem to have a similar flavor: take a relatively simple piece of geometry (Heron's formula, Pappus chains, etc) and then elaborate it into a page or two of algebra to get the formula.

1105:

If readers don't understand that an equation involving four things can be solved to find one of the things from the other three, then perhaps their level of mathematical sophistication does not reach that of the target audience for this article.

2050:#3b, requiring good articles to stay focused on their main topic and not go into unnecessary detail. This article had started to spread out into lots of topics that are not actually the formula for curvature for four mutually tangent circles. 1228:

It might be nice to add a section about more general relationships between circles, beyond Steiner chains. For example here's a paper where out of four circles two of them intersect, with the other two tangent to both of those and each-other:

2084:

The problem is that not every reader has quite as much experience or context as you have. So parts that seem to you to be unnecessary filler may be essential context for someone else. (This is of course a universal challenge in technical

3358:

History: Can't find the claim that Descartes' reasoning was "lost" in Coxeter, just that there is no complete proof in the letter. (From a quick skim read of the French original letter, it looks like there is very little proof there, if

1310:, which is exactly about finding tangent circles to triples of circles that are not necessarily tangent? Or, for constructing patterns of tangent circles with arbitrary planar graphs of adjacencies (like the golden window example), see 629: 3615:

Thank you! I didn't stop to think why it is clear the term under the square root can't be negative. In any case this means the formula is always valid without further qualifications, so there is probably no need to do anything here.

2194:

I'm not able to edit the article right now due to Knowledge technical problems (keeps timing out), but I made an image showing the 4 radii in the Heron's formula proof. If I can't successfully add it to the article, feel free to.

324:

of Descartes' Theorem. Ideally, at least the gist should appear in the lede, and the full statement should have been given by the next section (if not within the lede). A reader should not, for example, have first to read of the

973:; the convention of the "principal root" being the one with a positive real part doesn't seem to help. Although checking for geometric tangency seems the straightforward answer, I still hope to discover a more elementary test. 1002:

In this matter, for myself. I'd choose the later. Good grief, Descartes' theorem has it's own disambiguation page. How sucky is that? If he'd been any more productive, perhaps the thing to do would be to assign his theorems

1075:

It looks like an error: you start with three circles, solve the equation and reach a fourth. The way it's stated now reads like "you start with four circles, solve and equation, and reach a fourth circle", which doesn't make

2066:. It's a topic that is often useful in algorithms research, and so I would like to refer to our article when I want to look up the precise form of the bound. The form I want is always one of the forms in sub-subsection 350:

The article mentions how the theorem has been extended to spheres and even arbitrary dimension hyperspheres. Is it possible anyone can add the equations for either of those? Even just 3D spheres for now would be nice.

3930:

Re Gosset: added "the following year" and mentioned that he was one of several to do so, but again the poem is likely under copyright. Added the original version of the Gardner ref (easier to find for those with JSTOR

3459: 2796: 2730: 2664: 2598: 3397:

Integer curvatures: Just to clarify, is the root quadruple the one with the smallest signed curvature or the smallest positive curvature? (I.e. is everything inside one bounding circle? The image certainly suggests

1826:

Oriented projective geometry is symmetric with respect to point-line duality, so what difference does it make which ones you call the primitive objects? Or is this some kind of geometry that is not projective?

1632:(I guess with the signs I wrote above, these coefficients are the negatives of inversive distance as defined on Knowledge. The cosine of the angle between between two oriented lines is 1 for parallel lines, 1675:

for perpendicular lines. The negative inversive distance is a generalization which applies to oriented circles, and is the cosine of the angle between circles which intersect, or falls outside the interval

2102:

I wonder if we can rewrite the "proof" part to be a bit clearer (whether by reorganizing, adding a picture, chopping some parts, ...?) I find my eyes glaze over unless I really try to pay attention to it.

2877:

Meaning, is there a straightedge and compass construction of the fourth circle tangent to three given circles? Yes, but this is a different topic (and a much older one) than Descartes' theorem. It is the

1583:{\displaystyle Q={\begin{bmatrix}{\phantom {-}}1&-1&-1&-1\\-1&{\phantom {-}}1&-1&-1\\-1&-1&{\phantom {-}}1&-1\\-1&-1&-1&{\phantom {-}}1\\\end{bmatrix}}.} 1120:

This is not what I claimed readers would misunderstand. My comment was regarding the phrasing in plain English, which is a bit confusing, not the mathematical sophistication of the theorem itself.

4130:

important to the article, and is linked from the footnotes for any reader curious to read it. We could plausibly write a new prose statement of the theorem, but I doubt it would be any clearer. –

4237:, I think our remaining difference of opinion is not actually something that should stop the article from passing, so I will do that now. I guess one can argue that the article does not violate 4009:

various ways of defining it and their relations, etc.; I don't consider myself any kind of expert in this topic, and I don't know any particularly clear descriptions in the academic literature.

194: 2373: 3545: 2450: 2295:: maybe the negative radii are too tricksy, but I think signed areas are an improvement, because there are actually several possibilities here: (1) the 4th curvature is negative and 965:

is really just an extension of the problem of choosing a square root of a complex number. To rephrase my original problem more aptly, I haven't found a way to match a solution for k

3593: 4332: 259: 1984: 3974: 3854: 3234: 2009: 4395: 1841:

We are talking about a kind of dual metrical geometry, not just projective relationships. If you take lines to be primitives there is a whole set of transformations dual to

1809:'s dual approach to looking at Euclidean geometry by starting from oriented lines as the primitive object rather than points. I collected a long list of relevant sources at 4214:

and haven't appeared within the current or previous printed page.) But I'd probably recommend starting by fixing the crummy new default skin to improve rendering on screen.

2246: 3362:

Gosset extended the theorem and poem: would be nice to be told that is was just a year later (1937) and you could consider citing the relevant lines of the poem somewhere.

1368: 1206:

That's a bummer. We should perhaps at least track down some citations to the original sources then, so readers can save some effort on their path to disappointment. :-) –

2994: 3063: 3028: 1622: 4070:: remaining points are mainly whether to use the poem in the body and a few possible clarifications that should not take too long to resolve. Additionally there is a 4004: 3095: 2532: 2504: 2477: 2400: 2320: 818: 791: 764: 737: 710: 683: 656: 2862:

Is there a solution (straightedge and compass construction) using any right-angled triangle without first having to calculate the position of the Descartes circles?

896:, implies 2nd order contact. This is one order beyond tangency. Two osculating circles would be the same circle, since their curvature would be the same. It seems 3401:

Arbitrary four-circle configurations: perhaps spend a word or two explaining what +1 and -1 mean as "relative orientation between the ith and jth oriented circles".

3267: 1653: 2896:

Thanks for your quick answer. The question was, is there a construction with any right-angled triangle without first calculating the position of the 4th circle?--

1673: 1388: 1709: 301:

Is "curvature" the right term for ±1/r? I've seen explanations with curvature as always positive, 1/r, and "bend" used for the positive or negative curvature.

848:

there are three more solutions: each of the given three circles is itself mutually tangent to all three circles as well. So anyway, yes, existence is known. —

487: 378:

It says there is no 3-dimensional analogue of the complex numbers that the higher-dimensional cases can use. What about quaternions in a 3-dimensional case?

4155:

This binary distinction between footnotes that are just extended body text and footnotes that are purely citations to references is not supported by reality.

3257: 1154:

they may find proofs of the given facts, solutions and equations. Not implying that the article needs to contain proofs, since Knowledge doesn't do that.

4518: 4386: 184: 1864:

There are some nifty solutions to Euclidean problems using Laguerre geometry. For instance, given 3 oriented circles there are only 2 solutions to the

2062:

More generally, there is certainly a benefit to the reader in not distracting them with irrelevancies. A particularly bad example in that respect is

4523: 3239: 3937:

Jacobolus has explained already the sign of the square root. I'm not sure there's much we can expand on this in the article itself without sources.

3109:

The special case in which the three initial circles are tangent, as in the GeoGebra link, is easier than the general problem. See my 1999 web page

1790:

involved it's not properly a type of "distance" per se. I think they also messed up the sign; the product of a circle with itself should be 1 here.

4513: 160: 3407: 922:

They are certainly not osculating circles. But Soddy used the word "kissing" and for that reason I believe it should remain in the article. —

255: 3121:

2001). The right angle helps a tiny bit more, as two of the first three lines of the construction are already sides of the right triangle. —

2735: 2669: 2603: 2537: 2811:

I think this article should be readable by people who don't care to understand signed areas. And they are completely unnecessary here. See

2012:

few years out of college who only ever took one introductory linear algebra course), because the section then becomes much less concrete. –

387: 999:

It's one thing to follow historical norms (however horrible), it's another thing to have page titles that are immediately self-evident.

845: 386:

Testing the combination k_i = 1 1 1 gives 3 - 2*Sqrt(3) for the inner circle, which is negative. Is there supposed to be an initial +-?

4112: 907: 46: 32: 4508: 151: 112: 3391:

s are a bit distracting. Abbreviate as "eq." or take the word "equation" outside of the template so it is not in blue and boldface?

4503: 2944: 2046:

To me the whole article felt unfocused; not something I would feel comfortable nominating for GA status, particularly because of

1797:

A bit of a tangent here, while we're talking about oriented circles, I want to eventually write an article called something like

442: 426:

considered mutually tangent to a pair of circles also becomes questionable, as tangency implies having common points. Thoughts?

3652: 3262: 982: 952: 38: 4151:

Knowledge talk:Citing sources § Splitting notes is an arbitrary Knowledge-made standard with unknown consequences for readers

3371:"killer" problem: Not a fan of the one sentence paragraph, and while it is interesting trivia, it is a bit out of place here. 367: 3213:

Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

4266:

Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

4150: 1810: 3330:

Thanks! Don't feel like you need to rush; I am likely to be a bit busy with real-world work and slow to respond myself. —

3285: 87: 53: 4482: 4293: 3895:

45 I am wondering whether it is helpful to say when Caratheodory originally published this (he was long dead in 1954).

3306: 2123: 2116: 2089: 1328:

Let me be more precise. Lagarias/Mallows/Wilks found that Descartes' theorem could be expressed as a matrix equation

862:

That's all good info; this article should include enough of it to say that "such a circle must exist" and link where

3892:

44, 45 and similar could be separate footnotes instead of mixed with the citations, but that is definitely optional.

1159: 871: 2099:

We should add an image or two back to the relevant section of this article. Any advice on what it should include?

1771:

article has an incompatible definition of inversive distance using logs. The one I used here is the same as our

402: 4443: 4408: 4313: 4168: 4120: 4097: 4032: 3947: 3374:

Solving the quadratic: is there any reason the term under the square root should be positive? What if it isn't?

3335: 3126: 2916: 2887: 2820: 2325: 2277: 2167: 2162:

If we can find facsimiles of Descartes' original letter, Yamaji's work, or the 1824 Sangaku, those would fit. —

2139: 2075: 1832: 1780: 1319: 1197: 1111: 1052: 927: 853: 465: 4178:

detail notes" section. I think it would be a mistake to force a uniform format choice across the project (cf.

3494: 2405: 2186: 1928:

I'm thinking of adding something after the Steiner chain section along the lines of the following. Thoughts? –

1842: 658:

is zero, then the other circles must be external (you can't have a straight line fitting inside a circle), so

456: 391: 4356: 3924:

Re including more of the poem: I think it is still under copyright, so extended quotes would be problematic.

1857:. In the hyperbolic plane, Laguerre transformations are represented by fractional linear transformations of 1846: 911: 410: 3281: 1311: 1004: 438: 3550: 3465:

A nice article! I will comment on the GA criteria and have a look at some more of the sources in a bit. —

20: 3381: 2955: 2879: 1865: 1307: 1155: 1125: 1096: 1081: 867: 839:

It's a lot easier than that if you ignore the formula and think about it more geometrically. Perform an

231: 93: 2190:

Descartes' theorem can be proved from applying Heron's formula to 3 small triangles inside a larger one

2096:

into there, and that does give the subject a bit more room to expand without getting out of scope here.

1798: 137: 1957: 1121: 1092: 1077: 1061:

Yes, of course. It was request from one of russian users and I did not recognize answer immeiately...

1047:

No. The quadratic equation is the one labeled (1). It involves the radii of four circles, not three. —

820:. So I think Descartes' theorem does imply existence of a fourth circle. But what I'm doing here is 3162: 3144: 3135:

Thanks for the good solution! Maybe I will also draw your construction and enter it into the article

2940: 2932: 2901: 2867: 903: 434: 430: 355: 3110: 4460: 4439: 4416: 4404: 4370: 4349: 4310: 4234: 4221: 4186: 4164: 4134: 4116: 4108: 4093: 4067: 4050: 4028: 4017: 3943: 3607: 3331: 3122: 3101: 2912: 2883: 2846: 2834: 2816: 2812: 2802: 2292: 2273: 2264: 2252: 2199: 2163: 2154: 2135: 2107: 2071: 2037: 2030: 2016: 1945: 1932: 1913: 1906: 1902: 1885: 1828: 1818: 1776: 1759: 1315: 1297: 1210: 1193: 1179: 1171: 1107: 1048: 978: 948: 923: 863: 849: 461: 359: 335: 306: 284: 247: 3154: 1989: 1898: 159:

on Knowledge. If you would like to participate, please visit the project page, where you can join

2055: 1772: 1741: 1723: 1625: 1251: 840: 829: 406: 363: 288: 143: 3886:

16 could be used to clarify that Gosset was not the only one to generalise to higher dimensions.

3637: 3483:– I think it's fine to skip 'signed' in the lead, which is explained clearly enough in context. 2600:

still holds. The most complicated is to treat every case separately. In case (1) the formula is

2211: 631:, it's reasonable to ask if the expression under the square root can be negative. Certainly if 480:

There seems to be an implicit assertion that given three mutually tangent circles, there always

215: 127: 106: 3883:

15a doesn't really say that it is now called "Soddy's hexlet"; perhaps use 47 again as in 15b?

2805: 2202: 2051: 1331: 3600: 3366: 1858: 893: 42: 739:

is strictly negative, the first two circles must be small enough to fit inside the third, so

2961: 1282: 1243: 1066: 1037: 846:

Problem of Apollonius#Mutually tangent given circles: Soddy's circles and Descartes' theorem

3033: 2998: 1871:

Pretty well every feature of point-based metrical geometry has an oriented-line-based dual.

1597: 4478: 4428: 4246: 4200: 4083: 3982: 3962: 3908: 3869: 3621: 3470: 3320: 3300: 3197: 3158: 3140: 3068: 2936: 2925:

Maybe the following picture is better than my bad English. Is there a reference for this:

2897: 2863: 2510: 2482: 2455: 2378: 2298: 1806: 796: 769: 742: 715: 688: 661: 634: 1794:

leaves out consideration of intersecting circles, both of which seem to me like mistakes.

624:{\displaystyle k_{4}=k_{1}+k_{2}+k_{3}\pm 2{\sqrt {k_{1}k_{2}+k_{2}k_{3}+k_{3}k_{1}}}.\,} 3351:

Lead: The "bends" are signed inverse radii; not sure whether this detail should be here.

1267: 4432: 4412: 4218: 4183: 4131: 4059: 4047: 4014: 3819: 3765: 3604: 3098: 2843: 2831: 2799: 2261: 2249: 2196: 2151: 2104: 2067: 2063: 2047: 2034: 2013: 1942: 1929: 1910: 1882: 1876: 1815: 1756: 1635: 1294: 1207: 1176: 974: 944: 330: 302: 4425:. DYK is currently in unreviewed backlog mode and nominator has 182 past nominations. 2958:. This is a particular special case. The three circles at the vertices have diameters 1658: 1373: 4497: 4111:

summary of the theorem itself. I've started a discussion thread asking about this at

4075: 4071: 3791: 3669: 3642: 3487:– it might just be better to say that Descartes didn't fully describe his reasoning. 2248:

etc. instead of splitting each side up into two parts and labeling them separately. –

1802: 1768: 1679: 1170:

vague. The "Proof from Heron's formula" is algebraic and not especially satisfying. @

825: 4278:

The following is an archived discussion of the DYK nomination of the article below.

3114: 3778: 3724: 3710: 3696: 2534:

are differences of radii in cases (1–2), but to allow signed areas so that the sum

1247: 821: 1594:

What Kocik's paper claims is that these coefficients come from inverting a matrix

236: 3927:

Re Descartes' labors lost: reworded as "Descartes did not provide the reasoning".

2054:

is one of them. We have an article about it; we don't need to replicate it here.

2208:

I wonder if it would be better to show the whole sides of the triangles labeled

1854: 1062: 1033: 1008: 156: 4487: 4465: 4447: 4353:

in geometry came from a too-difficult mathematics problem posed to a princess?

4316: 4284:

Subsequent comments should be made on the appropriate discussion page (such as

4250: 4224: 4204: 4189: 4172: 4137: 4124: 4101: 4087: 4053: 4036: 4020: 3966: 3951: 3912: 3873: 3824: 3625: 3610: 3474: 3339: 3324: 3310: 3201: 3166: 3148: 3130: 3104: 2948: 2920: 2905: 2891: 2871: 2849: 2837: 2824: 2281: 2267: 2255: 2171: 2157: 2143: 2110: 2079: 2040: 2019: 1948: 1935: 1916: 1888: 1836: 1821: 1784: 1762: 1323: 1300: 1213: 1201: 1182: 1163: 1129: 1115: 1100: 1085: 1070: 1056: 1041: 1011: 986: 956: 931: 915: 875: 857: 833: 469: 446: 414: 395: 371: 340: 310: 291: 4474: 4242: 4196: 4113:

Knowledge:Media copyright questions § How much of copyrighted poem ok to copy?

4079: 4043: 3958: 3904: 3865: 3617: 3466: 3461:

isn't this worth attributing in text? Similar for some of the generalisations.

3316: 3296: 3193: 3153:

Thank you again for your references! I was able to use it well in the article

2260:

Okay I made that change, which I think makes the picture a bit more legible. –

2127: 2093: 251:

in geometry came from a too-difficult mathematics problem posed to a princess?

133: 1286: 1261:

And here is a paper with a generalization to arbitrary circles in the plane:

4374:, inspired several other geometer-poets to extend the poem and the theorem? 3682: 900:

should be removed and only mutual tangency should be given as a condition.

226: 4376:

Source: Martin Gardner, "Circles and spheres, and how they kiss and pack",

3288:. The edit link for this section can be used to add comments to the review. 1007:

numbers. At least then we could tell them apart on surface inspection. —

3850:

Article scope and focus are fine. No neutrality/stability issues detected.

4238: 3454:{\displaystyle \mathbf {k} ^{\mathsf {T}}\mathbf {Q} ^{-1}\mathbf {k} =0} 4377: 4195:

it exists). I did say I consider this "optional" above, and I mean it. —

3368:, Gosset was not the only one to write an n-dimensional poem about this. 2119:, I think, but that still identifies the variables used in that section. 4179: 2791:{\displaystyle \Delta _{123}+\Delta _{423}+\Delta _{143}=\Delta _{124}} 2725:{\displaystyle \Delta _{123}+\Delta _{423}=\Delta _{143}+\Delta _{124}} 2659:{\displaystyle \Delta _{123}=\Delta _{423}+\Delta _{143}+\Delta _{124}} 2593:{\displaystyle \Delta _{123}=\Delta _{423}+\Delta _{143}+\Delta _{124}} 1254: 1234:

Fox, M. D. (1980), "Formulae for the curvatures of circles in chains",

824:; is there a reference that clearly deals with the existence question? 3853:

Images are fine and relevant, with the only possible question whether

2926: 484:

a fourth—but this isn't clearly spelled out. Given a formula such as

4296:), unless there is consensus to re-open the discussion at this page. 4042:(I don't need any awards. I'm just happy to see articles improved.) @ 3489:

is there any reason the term under the square root should be positive

3394:

In the equation for z4, the roots are now complex; worth mentioning?

1775:

article (and the source I used), but that confused me for a while. —

996:

An external resource bears the title "Descartes' circle theorem".

3387:

template linking to the equations is nice, but the many bold faced

1746: 401:

It is outer circe. Inner will be 3+2*Sqrt(3). You can also look at

3136: 2185: 1897:

Oh, from looking at that list again, Coolidge has the sign right (

1738:

Proof of Descartes circle formula and its generalization clarified

1728: 4160:

Jacobolus found and added an orig-year for the Caratheodory cite.

3111:

https://ics.uci.edu/~eppstein/junkyard/tangencies/apollonian.html

2150:

Also are there any images we can find for the history section? –

3818:

Note: this represents where the article stands relative to the

2842:

I guess your alternative is okay. Let me read more carefully. –

961:

Thinking more about it, I suppose the ± in the expression for z

320:

This article takes entirely too long before it states even the

1941:

Removing this draft subsection and moving it to the article. –

210: 69: 15: 2092:

instead of just deleting that part; we should probably merge

4355:

Source: Dana Mackenzie, "The Princess and the Philosopher",

265: 3844:

Lead is perhaps a bit short, other than including the poem.

3315:

Planning to review, should not take more than a few days. —

1655:

for lines pointed in diametrically opposite directions, or

1091:

construct a fourth circle tangent to all of them.", maybe?

3918:

A few replies, not intended to cover the entire review:

4435:

the nomination until the hook appears on the Main Page.

4337: 4328: 2452:

but outside the triangle across the opposite side, (3)

1306:

Isn't that really more suitable for our article on the

892:

circles. Yet the standard definition of kissing, i.e.

59: 3553: 3497: 1410: 258:. The nomination discussion and review may be seen at 4364:: ... that the publication of Frederick Soddy's poem 3985: 3410: 3071: 3036: 3001: 2964: 2738: 2672: 2606: 2540: 2513: 2485: 2458: 2408: 2381: 2328: 2301: 2214: 2068:

Chernoff bound § Multiplicative form (relative error)

1992: 1960: 1682: 1661: 1638: 1600: 1398: 1376: 1334: 799: 772: 745: 718: 691: 664: 637: 490: 3975:

File:Generalized_circles_in_the_hyperbolic_plane.png

3855:

File:Generalized_circles_in_the_hyperbolic_plane.png

2830:

proof is to just enumerate the possibilities then. –

2059:

sub-sub-section I removed is a tangent to a tangent.

1877:

http://fractal.dam.fmph.uniba.sk/~chalmo/GM/edge.pdf

793:

are both positive and bigger in absolute value than

260:

Template:Did you know nominations/Descartes' theorem

155:, a collaborative effort to improve the coverage of 4163:

I think that mainly leaves the lead length query. —

3603:

a "root quadruple" includes the bounding circle. –

2115:Re image: Something less busy than the ones now in 1875:property preserved under Laguerre transformations. 1714:Kocik has some other papers with more detail, e.g. 3998: 3587: 3539: 3453: 3089: 3057: 3022: 2988: 2790: 2724: 2658: 2592: 2526: 2498: 2479:is at infinity, (4) the curvature is positive and 2471: 2444: 2394: 2367: 2314: 2240: 2003: 1978: 1811:Talk:Laguerre transformations § List of references 1703: 1667: 1647: 1616: 1582: 1382: 1362: 1224:generalizations to not-necessarily-tangent circles 812: 785: 758: 731: 704: 677: 650: 623: 4092:Dropped. More when I have more time to respond. — 4046:thanks for taking the time to make this review. – 1954:I think it's helpful to directly show the matrix 4396:Template:Did you know nominations/Anders Åkerman 3595:I'm not sure how helpful it is to discuss this. 1909:) for the cosine of the angle between circles. – 45:. If it no longer meets these criteria, you can 1390:is the vector of curvatures of the circles and 1187:To be honest, the Heron's formula proof is the 4074:, namely "unfortunately", which runs afoul of 8: 4298:No further edits should be made to this page 287:should be a disambiguation page? Thoughts? 4387:Template:Did you know nominations/Rita Cox 3921:Re gloss of bends in lead: added "signed". 3217: 2930: 2368:{\displaystyle \triangle P_{1}P_{2}P_{3},} 888:In its introduction the article refers to 101: 3990: 3984: 3572: 3552: 3540:{\textstyle 2\cdot 2-2\cdot 1-2\cdot 1=0} 3496: 3440: 3431: 3426: 3418: 3417: 3412: 3409: 3070: 3035: 3000: 2963: 2782: 2769: 2756: 2743: 2737: 2716: 2703: 2690: 2677: 2671: 2650: 2637: 2624: 2611: 2605: 2584: 2571: 2558: 2545: 2539: 2518: 2512: 2490: 2484: 2463: 2457: 2445:{\displaystyle \triangle P_{1}P_{2}P_{3}} 2436: 2426: 2416: 2407: 2386: 2380: 2356: 2346: 2336: 2327: 2306: 2300: 2232: 2219: 2213: 1993: 1991: 1967: 1962: 1959: 1745: 1727: 1681: 1660: 1637: 1605: 1599: 1558: 1557: 1510: 1509: 1462: 1461: 1414: 1413: 1405: 1397: 1375: 1339: 1333: 804: 798: 777: 771: 750: 744: 723: 717: 696: 690: 669: 663: 642: 636: 608: 598: 585: 575: 562: 552: 546: 534: 521: 508: 495: 489: 239:). The text of the entry was as follows: 3097:), so those are not hard to construct. – 1150:The article could usefully tell readers 346:Equations for spheres too? Quaternions? 4063:a question of getting a reward sticker. 3248: 3220: 619: 103: 4154: 3596: 3488: 3484: 3480: 3419: 3365:Actually, according to Martin Gardner 2858:Straightedge and compass construction 1805:that does a better job of describing 1629:is a scalar multiple of its inverse.) 969:with the corresponding solution for z 256:Knowledge:Recent additions/2024/April 254:A record of the entry may be seen at 7: 3647: 3588:{\textstyle 2+2-1\pm {\sqrt {0}}=3.} 3209:The following discussion is closed. 3115:Tangent Spheres and Triangle Centers 1711:for circles which do not intersect.) 149:This article is within the scope of 75: 73: 4403:Improved to Good Article status by 3547:and the two other "bends" are both 3139:with the two links as references.-- 2130:: yes, this looks reasonable to me. 92:It is of interest to the following 3674: 2779: 2766: 2753: 2740: 2713: 2700: 2687: 2674: 2647: 2634: 2621: 2608: 2581: 2568: 2555: 2542: 2409: 2375:(2) the curvature is negative and 2329: 2272:New picture looks helpful to me. — 1720:A theorem on circle configurations 14: 4519:Low-priority mathematics articles 3377:Locating the circle centers: The 2402:is in the inside of one angle of 1979:{\displaystyle \mathbf {Q} ^{-1}} 1236:The American Mathematical Monthly 266: 169:Knowledge:WikiProject Mathematics 41:. If you can improve it further, 4464: 4262:The discussion above is closed. 3864:Just source checks left to do. — 3823: 3805: 3795: 3782: 3769: 3756: 3746: 3728: 3714: 3700: 3686: 3673: 3660: 3634:General comments and GA criteria 3441: 3427: 3413: 2025:special case of "steiner chains" 1994: 1963: 214: 172:Template:WikiProject Mathematics 136: 126: 105: 74: 19: 4524:Knowledge Did you know articles 4281:Please do not modify this page. 1814:conventions. Any suggestions? – 189:This article has been rated as 3860:All free and nicely captioned. 1698: 1683: 1248:10.1080/00029890.1980.11995129 1175:this "good article" quality. – 29:has been listed as one of the 1: 4514:GA-Class mathematics articles 3167:11:46, 18 December 2023 (UTC) 3149:15:19, 14 December 2023 (UTC) 3131:06:42, 14 December 2023 (UTC) 3105:06:36, 14 December 2023 (UTC) 2949:06:04, 14 December 2023 (UTC) 2921:02:08, 14 December 2023 (UTC) 2906:22:21, 13 December 2023 (UTC) 2892:21:59, 13 December 2023 (UTC) 2872:21:49, 13 December 2023 (UTC) 2181: 2088:Thanks for at least creating 2004:{\displaystyle \mathbf {k} ,} 1146:Help those who would prove it 1057:07:51, 19 February 2018 (UTC) 1042:07:33, 19 February 2018 (UTC) 396:08:23, 16 December 2009 (UTC) 372:06:03, 26 February 2009 (UTC) 311:14:29, 13 February 2008 (UTC) 163:and see a list of open tasks. 4427:Post-promotion hook changes 4394:: 2nd QPQ for backlog mode: 3806: 3796: 3783: 3770: 3757: 3747: 3741: 3729: 3715: 3701: 3687: 3661: 3639: 1164:02:58, 25 January 2021 (UTC) 1130:04:01, 14 October 2018 (UTC) 1116:03:26, 14 October 2018 (UTC) 1101:02:46, 14 October 2018 (UTC) 1086:02:41, 14 October 2018 (UTC) 1012:18:45, 15 January 2017 (UTC) 876:02:54, 25 January 2021 (UTC) 341:03:26, 2 November 2008 (UTC) 4431:on the talk page; consider 4294:Knowledge talk:Did you know 4286:this nomination's talk page 3640: 3286:Talk:Descartes' theorem/GA1 2241:{\displaystyle r_{1}+r_{2}} 2124:Soddy circles of a triangle 2117:Soddy circles of a triangle 2090:Soddy circles of a triangle 1017:that for every four kissing 470:00:58, 14 August 2010 (UTC) 447:16:02, 13 August 2010 (UTC) 415:18:04, 15 August 2010 (UTC) 4540: 4488:21:20, 26 March 2024 (UTC) 4448:07:25, 25 March 2024 (UTC) 4347:... that the discovery of 4317:01:50, 14 April 2024 (UTC) 4251:17:47, 23 March 2024 (UTC) 4225:00:21, 20 March 2024 (UTC) 4205:21:00, 19 March 2024 (UTC) 4190:13:59, 19 March 2024 (UTC) 4173:07:14, 19 March 2024 (UTC) 4138:07:09, 19 March 2024 (UTC) 4125:06:38, 19 March 2024 (UTC) 4102:15:51, 18 March 2024 (UTC) 4088:09:23, 18 March 2024 (UTC) 4054:20:51, 17 March 2024 (UTC) 4037:20:29, 17 March 2024 (UTC) 4021:17:19, 17 March 2024 (UTC) 3967:18:11, 17 March 2024 (UTC) 3952:01:50, 17 March 2024 (UTC) 3913:10:28, 16 March 2024 (UTC) 3874:08:02, 16 March 2024 (UTC) 3626:18:19, 17 March 2024 (UTC) 3611:15:06, 16 March 2024 (UTC) 3475:07:30, 16 March 2024 (UTC) 3340:22:26, 11 March 2024 (UTC) 3325:21:17, 11 March 2024 (UTC) 3311:21:17, 11 March 2024 (UTC) 3202:17:47, 23 March 2024 (UTC) 1363:{\displaystyle k^{T}Qk=0,} 987:22:49, 18 March 2015 (UTC) 957:21:03, 18 March 2015 (UTC) 932:16:07, 21 March 2014 (UTC) 916:19:50, 20 March 2014 (UTC) 712:will be positive. And if 283:I can't help wondering of 245:... that the discovery of 4509:Mathematics good articles 4421:Number of QPQs required: 3065:(for a triangle of sides 2850:20:49, 21 July 2023 (UTC) 2838:20:48, 21 July 2023 (UTC) 2825:20:23, 21 July 2023 (UTC) 2806:20:10, 21 July 2023 (UTC) 2282:07:28, 21 July 2023 (UTC) 2268:00:46, 21 July 2023 (UTC) 2256:00:29, 21 July 2023 (UTC) 2203:00:20, 21 July 2023 (UTC) 2172:18:03, 20 July 2023 (UTC) 2158:07:25, 20 July 2023 (UTC) 2144:18:02, 20 July 2023 (UTC) 2111:07:22, 20 July 2023 (UTC) 2080:05:00, 20 July 2023 (UTC) 2041:01:58, 20 July 2023 (UTC) 2020:03:19, 19 June 2023 (UTC) 1949:22:05, 15 June 2023 (UTC) 1936:20:51, 12 June 2023 (UTC) 1917:02:55, 12 June 2023 (UTC) 1889:05:29, 12 June 2023 (UTC) 1837:03:45, 12 June 2023 (UTC) 1822:02:38, 12 June 2023 (UTC) 1785:01:59, 12 June 2023 (UTC) 1763:00:33, 12 June 2023 (UTC) 1324:07:17, 11 June 2023 (UTC) 1301:07:02, 11 June 2023 (UTC) 1071:13:31, 5 March 2018 (UTC) 938:Correspondence of ± signs 858:06:48, 24 June 2011 (UTC) 834:04:19, 24 June 2011 (UTC) 421:Clarify in Special Cases? 382:Equation needs clarifying 235:column on 23 April 2024 ( 188: 121: 100: 33:Mathematics good articles 4264:Please do not modify it. 3346:Content and prose review 3211:Please do not modify it. 1847:Laguerre transformations 1287:10.4169/002557010X529815 1214:18:59, 8 July 2023 (UTC) 1202:18:32, 8 July 2023 (UTC) 1183:18:15, 8 July 2023 (UTC) 292:22:35, 23 May 2005 (UTC) 225:appeared on Knowledge's 195:project's priority scale 4504:Knowledge good articles 4290:the article's talk page 4271:Did you know nomination 4153:and my comments there " 3934:Killer problem removed. 2989:{\displaystyle -a+b+c,} 152:WikiProject Mathematics 4078:. Better to drop it. — 4000: 3841:No major prose issues. 3589: 3541: 3455: 3091: 3059: 3058:{\displaystyle a+b-c,} 3024: 3023:{\displaystyle a-b+c,} 2990: 2792: 2726: 2660: 2594: 2528: 2500: 2473: 2446: 2396: 2369: 2316: 2242: 2191: 2005: 1980: 1843:Möbius transformations 1705: 1669: 1649: 1624:whose entries are the 1618: 1617:{\displaystyle Q^{-1}} 1584: 1384: 1364: 1312:circle packing theorem 1005:Bach-Werke-Verzeichnis 992:Possible title upgrade 814: 787: 760: 733: 706: 679: 652: 625: 271: 82:This article is rated 4001: 3999:{\displaystyle k_{i}} 3820:Good Article criteria 3792:free or tagged images 3590: 3542: 3456: 3137:Apollonisches Problem 3092: 3090:{\displaystyle a,b,c} 3060: 3025: 2991: 2956:Problem of Apollonius 2880:Problem of Apollonius 2793: 2732:and in case (4) e.g. 2727: 2661: 2595: 2529: 2527:{\displaystyle P_{4}} 2501: 2499:{\displaystyle P_{4}} 2474: 2472:{\displaystyle P_{4}} 2447: 2397: 2395:{\displaystyle P_{4}} 2370: 2317: 2315:{\displaystyle P_{4}} 2243: 2189: 2182:Heron's formula proof 2006: 1981: 1866:problem of Apollonius 1736:Kocik, Jerzy (2019), 1718:Kocik, Jerzy (2007), 1706: 1670: 1650: 1619: 1585: 1385: 1365: 1308:Problem of Apollonius 1266:Kocik, Jerzy (2010), 815: 813:{\displaystyle k_{3}} 788: 786:{\displaystyle k_{2}} 761: 759:{\displaystyle k_{1}} 734: 732:{\displaystyle k_{3}} 707: 705:{\displaystyle k_{2}} 680: 678:{\displaystyle k_{1}} 653: 651:{\displaystyle k_{3}} 626: 457:Möbius transformation 269: 39:good article criteria 3983: 3551: 3495: 3481:signed inverse radii 3408: 3069: 3034: 2999: 2962: 2736: 2670: 2604: 2538: 2511: 2483: 2456: 2406: 2379: 2326: 2299: 2212: 1990: 1958: 1680: 1659: 1636: 1598: 1563: 1515: 1467: 1419: 1396: 1374: 1332: 1275:Mathematics Magazine 797: 770: 743: 716: 689: 662: 635: 488: 175:mathematics articles 4107:a readable and non- 3645:review progress box 3119:Amer. Math. Monthly 3113:and related paper " 2134:specific article. — 1626:inversive distances 1559: 1511: 1463: 1415: 864:User:David Eppstein 57:: March 23, 2024. ( 4371:Descartes' theorem 4350:Descartes' theorem 3996: 3822:. Criteria marked 3585: 3537: 3485:reasoning ... lost 3451: 3212: 3155:Satz von Descartes 3087: 3055: 3020: 2986: 2788: 2722: 2656: 2590: 2524: 2496: 2469: 2442: 2392: 2365: 2312: 2238: 2192: 2056:Inversive distance 2001: 1976: 1859:hyperbolic numbers 1773:inversive distance 1701: 1665: 1648:{\displaystyle -1} 1645: 1614: 1580: 1571: 1380: 1360: 985:) Yeah, that guy. 955:) Yeah, that guy. 810: 783: 756: 729: 702: 675: 648: 621: 620: 285:Descartes' theorem 272: 248:Descartes' theorem 223:Descartes' theorem 144:Mathematics portal 88:content assessment 27:Descartes' theorem 4486: 4379: 4358: 3838: 3837: 3834: 3833: 3830: 3601:Apollonian gasket 3577: 3276: 3275: 3210: 2951: 2935:comment added by 2666:in case (2) e.g. 2122:Re merge between 1799:Laguerre geometry 1668:{\displaystyle 0} 1383:{\displaystyle k} 906:comment added by 894:Osculating circle 822:original research 614: 450: 433:comment added by 403:wikibooks article 375: 358:comment added by 329:of the theorem. — 276: 275: 209: 208: 205: 204: 201: 200: 68: 67: 64: 4531: 4472: 4471: 4468: 4375: 4366:The Kiss Precise 4354: 4305:The result was: 4283: 4007: 4005: 4003: 4002: 3997: 3995: 3994: 3857:is helpful here. 3827: 3816: 3809: 3808: 3799: 3798: 3786: 3785: 3773: 3772: 3760: 3759: 3750: 3749: 3732: 3731: 3718: 3717: 3704: 3703: 3690: 3689: 3677: 3676: 3664: 3663: 3648: 3638: 3594: 3592: 3591: 3586: 3578: 3573: 3546: 3544: 3543: 3538: 3460: 3458: 3457: 3452: 3444: 3439: 3438: 3430: 3424: 3423: 3422: 3416: 3386: 3380: 3230:Copyvio detector 3218: 3157:. With regards-- 3096: 3094: 3093: 3088: 3064: 3062: 3061: 3056: 3029: 3027: 3026: 3021: 2995: 2993: 2992: 2987: 2797: 2795: 2794: 2789: 2787: 2786: 2774: 2773: 2761: 2760: 2748: 2747: 2731: 2729: 2728: 2723: 2721: 2720: 2708: 2707: 2695: 2694: 2682: 2681: 2665: 2663: 2662: 2657: 2655: 2654: 2642: 2641: 2629: 2628: 2616: 2615: 2599: 2597: 2596: 2591: 2589: 2588: 2576: 2575: 2563: 2562: 2550: 2549: 2533: 2531: 2530: 2525: 2523: 2522: 2505: 2503: 2502: 2497: 2495: 2494: 2478: 2476: 2475: 2470: 2468: 2467: 2451: 2449: 2448: 2443: 2441: 2440: 2431: 2430: 2421: 2420: 2401: 2399: 2398: 2393: 2391: 2390: 2374: 2372: 2371: 2366: 2361: 2360: 2351: 2350: 2341: 2340: 2321: 2319: 2318: 2313: 2311: 2310: 2247: 2245: 2244: 2239: 2237: 2236: 2224: 2223: 2052:Steiner's porism 2010: 2008: 2007: 2002: 1997: 1985: 1983: 1982: 1977: 1975: 1974: 1966: 1750: 1749: 1732: 1731: 1710: 1708: 1707: 1704:{\displaystyle } 1702: 1674: 1672: 1671: 1666: 1654: 1652: 1651: 1646: 1623: 1621: 1620: 1615: 1613: 1612: 1589: 1587: 1586: 1581: 1576: 1575: 1565: 1564: 1517: 1516: 1469: 1468: 1421: 1420: 1389: 1387: 1386: 1381: 1369: 1367: 1366: 1361: 1344: 1343: 1289: 1272: 1257: 918: 819: 817: 816: 811: 809: 808: 792: 790: 789: 784: 782: 781: 765: 763: 762: 757: 755: 754: 738: 736: 735: 730: 728: 727: 711: 709: 708: 703: 701: 700: 684: 682: 681: 676: 674: 673: 657: 655: 654: 649: 647: 646: 630: 628: 627: 622: 615: 613: 612: 603: 602: 590: 589: 580: 579: 567: 566: 557: 556: 547: 539: 538: 526: 525: 513: 512: 500: 499: 449: 427: 374: 352: 338: 268: 218: 211: 177: 176: 173: 170: 167: 146: 141: 140: 130: 123: 122: 117: 109: 102: 85: 79: 78: 77: 70: 62: 60:Reviewed version 51: 23: 16: 4539: 4538: 4534: 4533: 4532: 4530: 4529: 4528: 4494: 4493: 4492: 4469: 4436: 4344: 4342: 4338:Article history 4279: 4273: 4268: 4267: 3986: 3981: 3980: 3978: 3847:Layout is fine. 3804:pics relevant ( 3636: 3599:– judging from 3549: 3548: 3493: 3492: 3425: 3411: 3406: 3405: 3384: 3378: 3348: 3280:This review is 3272: 3244: 3215: 3206: 3205: 3204: 3187: 3067: 3066: 3032: 3031: 2997: 2996: 2960: 2959: 2860: 2778: 2765: 2752: 2739: 2734: 2733: 2712: 2699: 2686: 2673: 2668: 2667: 2646: 2633: 2620: 2607: 2602: 2601: 2580: 2567: 2554: 2541: 2536: 2535: 2514: 2509: 2508: 2486: 2481: 2480: 2459: 2454: 2453: 2432: 2422: 2412: 2404: 2403: 2382: 2377: 2376: 2352: 2342: 2332: 2324: 2323: 2302: 2297: 2296: 2228: 2215: 2210: 2209: 2184: 2027: 1988: 1987: 1961: 1956: 1955: 1807:Edmond Laguerre 1735: 1717: 1678: 1677: 1657: 1656: 1634: 1633: 1601: 1596: 1595: 1570: 1569: 1555: 1547: 1539: 1530: 1529: 1521: 1507: 1499: 1490: 1489: 1481: 1473: 1459: 1450: 1449: 1441: 1433: 1425: 1406: 1394: 1393: 1372: 1371: 1335: 1330: 1329: 1270: 1268:"Golden window" 1265: 1233: 1226: 1148: 1026:that for every 1019: 994: 972: 968: 964: 940: 901: 886: 800: 795: 794: 773: 768: 767: 746: 741: 740: 719: 714: 713: 692: 687: 686: 665: 660: 659: 638: 633: 632: 604: 594: 581: 571: 558: 548: 530: 517: 504: 491: 486: 485: 478: 460:not a circle. — 428: 423: 384: 353: 348: 339: 334: 318: 299: 281: 174: 171: 168: 165: 164: 142: 135: 115: 86:on Knowledge's 83: 58: 12: 11: 5: 4537: 4535: 4527: 4526: 4521: 4516: 4511: 4506: 4496: 4495: 4491: 4490: 4461:David Eppstein 4440:David Eppstein 4429:will be logged 4405:David Eppstein 4402: 4401: 4400: 4399: 4398: 4389: 4380: 4341: 4340: 4335: 4325: 4323: 4319: 4311:PrimalMustelid 4303: 4302: 4274: 4272: 4269: 4261: 4260: 4259: 4258: 4257: 4256: 4255: 4254: 4253: 4235:David Eppstein 4231: 4230: 4229: 4228: 4227: 4215: 4211: 4165:David Eppstein 4161: 4158: 4146: 4142: 4141: 4140: 4117:David Eppstein 4104: 4094:David Eppstein 4068:David Eppstein 4064: 4029:David Eppstein 4024: 4023: 4010: 3993: 3989: 3971: 3970: 3969: 3944:David Eppstein 3940: 3939: 3938: 3935: 3932: 3928: 3925: 3922: 3900: 3899: 3896: 3893: 3890: 3887: 3884: 3881: 3862: 3861: 3858: 3851: 3848: 3845: 3842: 3836: 3835: 3832: 3831: 3829: 3828:are unassessed 3813: 3812: 3740: 3739: 3736: 3735: 3635: 3632: 3631: 3630: 3629: 3628: 3597:root quadruple 3584: 3581: 3576: 3571: 3568: 3565: 3562: 3559: 3556: 3536: 3533: 3530: 3527: 3524: 3521: 3518: 3515: 3512: 3509: 3506: 3503: 3500: 3463: 3462: 3450: 3447: 3443: 3437: 3434: 3429: 3421: 3415: 3402: 3399: 3395: 3392: 3375: 3372: 3369: 3363: 3360: 3356: 3352: 3347: 3344: 3343: 3342: 3332:David Eppstein 3291: 3290: 3274: 3273: 3271: 3270: 3265: 3260: 3254: 3251: 3250: 3246: 3245: 3243: 3242: 3240:External links 3237: 3232: 3226: 3223: 3222: 3216: 3207: 3191: 3190: 3189: 3188: 3186: 3183: 3182: 3181: 3180: 3179: 3178: 3177: 3176: 3175: 3174: 3173: 3172: 3171: 3170: 3169: 3151: 3123:David Eppstein 3086: 3083: 3080: 3077: 3074: 3054: 3051: 3048: 3045: 3042: 3039: 3019: 3016: 3013: 3010: 3007: 3004: 2985: 2982: 2979: 2976: 2973: 2970: 2967: 2913:David Eppstein 2884:David Eppstein 2859: 2856: 2855: 2854: 2853: 2852: 2840: 2817:David Eppstein 2785: 2781: 2777: 2772: 2768: 2764: 2759: 2755: 2751: 2746: 2742: 2719: 2715: 2711: 2706: 2702: 2698: 2693: 2689: 2685: 2680: 2676: 2653: 2649: 2645: 2640: 2636: 2632: 2627: 2623: 2619: 2614: 2610: 2587: 2583: 2579: 2574: 2570: 2566: 2561: 2557: 2553: 2548: 2544: 2521: 2517: 2493: 2489: 2466: 2462: 2439: 2435: 2429: 2425: 2419: 2415: 2411: 2389: 2385: 2364: 2359: 2355: 2349: 2345: 2339: 2335: 2331: 2309: 2305: 2293:David Eppstein 2289: 2288: 2287: 2286: 2285: 2284: 2274:David Eppstein 2235: 2231: 2227: 2222: 2218: 2183: 2180: 2179: 2178: 2177: 2176: 2175: 2174: 2164:David Eppstein 2148: 2147: 2146: 2136:David Eppstein 2131: 2120: 2100: 2097: 2086: 2072:David Eppstein 2064:Chernoff bound 2060: 2031:David Eppstein 2026: 2023: 2000: 1996: 1973: 1970: 1965: 1952: 1951: 1926: 1925: 1924: 1923: 1922: 1921: 1920: 1919: 1895: 1894: 1893: 1892: 1891: 1879: 1872: 1869: 1862: 1850: 1829:David Eppstein 1795: 1791: 1777:David Eppstein 1753: 1752: 1751: 1733: 1712: 1700: 1697: 1694: 1691: 1688: 1685: 1664: 1644: 1641: 1630: 1611: 1608: 1604: 1592: 1591: 1590: 1579: 1574: 1568: 1562: 1556: 1554: 1551: 1548: 1546: 1543: 1540: 1538: 1535: 1532: 1531: 1528: 1525: 1522: 1520: 1514: 1508: 1506: 1503: 1500: 1498: 1495: 1492: 1491: 1488: 1485: 1482: 1480: 1477: 1474: 1472: 1466: 1460: 1458: 1455: 1452: 1451: 1448: 1445: 1442: 1440: 1437: 1434: 1432: 1429: 1426: 1424: 1418: 1412: 1411: 1409: 1404: 1401: 1379: 1359: 1356: 1353: 1350: 1347: 1342: 1338: 1316:David Eppstein 1291: 1290: 1281:(5): 384–390, 1259: 1258: 1242:(9): 708–715, 1225: 1222: 1221: 1220: 1219: 1218: 1217: 1216: 1194:David Eppstein 1172:David Eppstein 1147: 1144: 1143: 1142: 1141: 1140: 1139: 1138: 1137: 1136: 1135: 1134: 1133: 1132: 1108:David Eppstein 1049:David Eppstein 1021:Is it error? 1018: 1015: 993: 990: 970: 966: 962: 939: 936: 935: 934: 924:David Eppstein 885: 882: 881: 880: 879: 878: 850:David Eppstein 807: 803: 780: 776: 753: 749: 726: 722: 699: 695: 672: 668: 645: 641: 618: 611: 607: 601: 597: 593: 588: 584: 578: 574: 570: 565: 561: 555: 551: 545: 542: 537: 533: 529: 524: 520: 516: 511: 507: 503: 498: 494: 477: 474: 473: 472: 462:David Eppstein 422: 419: 418: 417: 388:121.45.179.128 383: 380: 347: 344: 333: 317: 314: 298: 295: 280: 279:Disambiguation 277: 274: 273: 263: 253: 252: 219: 207: 206: 203: 202: 199: 198: 187: 181: 180: 178: 161:the discussion 148: 147: 131: 119: 118: 110: 98: 97: 91: 80: 66: 65: 50: 24: 13: 10: 9: 6: 4: 3: 2: 4536: 4525: 4522: 4520: 4517: 4515: 4512: 4510: 4507: 4505: 4502: 4501: 4499: 4489: 4484: 4480: 4476: 4467: 4462: 4457: 4453: 4452: 4451: 4449: 4445: 4441: 4434: 4430: 4426: 4424: 4418: 4414: 4410: 4406: 4397: 4393: 4390: 4388: 4384: 4381: 4378: 4373: 4372: 4367: 4363: 4360: 4359: 4357: 4352: 4351: 4346: 4345: 4339: 4336: 4334: 4330: 4327: 4326: 4322: 4320: 4318: 4315: 4312: 4308: 4301: 4299: 4295: 4291: 4287: 4282: 4276: 4275: 4270: 4265: 4252: 4248: 4244: 4240: 4236: 4232: 4226: 4223: 4220: 4216: 4212: 4208: 4207: 4206: 4202: 4198: 4193: 4192: 4191: 4188: 4185: 4181: 4176: 4175: 4174: 4170: 4166: 4162: 4159: 4156: 4152: 4147: 4143: 4139: 4136: 4133: 4128: 4127: 4126: 4122: 4118: 4114: 4110: 4105: 4103: 4099: 4095: 4091: 4090: 4089: 4085: 4081: 4077: 4076:MOS:EDITORIAL 4073: 4069: 4065: 4061: 4057: 4056: 4055: 4052: 4049: 4045: 4041: 4040: 4039: 4038: 4034: 4030: 4022: 4019: 4016: 4011: 3991: 3987: 3976: 3972: 3968: 3964: 3960: 3955: 3954: 3953: 3949: 3945: 3941: 3936: 3933: 3929: 3926: 3923: 3920: 3919: 3917: 3916: 3915: 3914: 3910: 3906: 3897: 3894: 3891: 3888: 3885: 3882: 3879: 3878: 3877: 3875: 3871: 3867: 3859: 3856: 3852: 3849: 3846: 3843: 3840: 3839: 3826: 3821: 3817: 3815: 3814: 3811: 3801: 3793: 3788: 3780: 3775: 3767: 3762: 3752: 3742: 3738: 3737: 3734: 3726: 3720: 3712: 3706: 3698: 3692: 3684: 3679: 3671: 3666: 3656: 3654: 3650: 3649: 3646: 3644: 3633: 3627: 3623: 3619: 3614: 3613: 3612: 3609: 3606: 3602: 3598: 3582: 3579: 3574: 3569: 3566: 3563: 3560: 3557: 3554: 3534: 3531: 3528: 3525: 3522: 3519: 3516: 3513: 3510: 3507: 3504: 3501: 3498: 3490: 3486: 3482: 3479: 3478: 3477: 3476: 3472: 3468: 3448: 3445: 3435: 3432: 3404:The equation 3403: 3400: 3396: 3393: 3390: 3383: 3376: 3373: 3370: 3367: 3364: 3361: 3357: 3353: 3350: 3349: 3345: 3341: 3337: 3333: 3329: 3328: 3327: 3326: 3322: 3318: 3313: 3312: 3308: 3305: 3302: 3298: 3295: 3289: 3287: 3283: 3278: 3277: 3269: 3266: 3264: 3261: 3259: 3256: 3255: 3253: 3252: 3247: 3241: 3238: 3236: 3233: 3231: 3228: 3227: 3225: 3224: 3219: 3214: 3203: 3199: 3195: 3184: 3168: 3164: 3160: 3156: 3152: 3150: 3146: 3142: 3138: 3134: 3133: 3132: 3128: 3124: 3120: 3116: 3112: 3108: 3107: 3106: 3103: 3100: 3084: 3081: 3078: 3075: 3072: 3052: 3049: 3046: 3043: 3040: 3037: 3017: 3014: 3011: 3008: 3005: 3002: 2983: 2980: 2977: 2974: 2971: 2968: 2965: 2957: 2953: 2952: 2950: 2946: 2942: 2938: 2934: 2928: 2924: 2923: 2922: 2918: 2914: 2909: 2908: 2907: 2903: 2899: 2895: 2894: 2893: 2889: 2885: 2881: 2876: 2875: 2874: 2873: 2869: 2865: 2857: 2851: 2848: 2845: 2841: 2839: 2836: 2833: 2828: 2827: 2826: 2822: 2818: 2814: 2810: 2809: 2808: 2807: 2804: 2801: 2783: 2775: 2770: 2762: 2757: 2749: 2744: 2717: 2709: 2704: 2696: 2691: 2683: 2678: 2651: 2643: 2638: 2630: 2625: 2617: 2612: 2585: 2577: 2572: 2564: 2559: 2551: 2546: 2519: 2515: 2491: 2487: 2464: 2460: 2437: 2433: 2427: 2423: 2417: 2413: 2387: 2383: 2362: 2357: 2353: 2347: 2343: 2337: 2333: 2307: 2303: 2294: 2283: 2279: 2275: 2271: 2270: 2269: 2266: 2263: 2259: 2258: 2257: 2254: 2251: 2233: 2229: 2225: 2220: 2216: 2207: 2206: 2205: 2204: 2201: 2198: 2188: 2173: 2169: 2165: 2161: 2160: 2159: 2156: 2153: 2149: 2145: 2141: 2137: 2132: 2129: 2125: 2121: 2118: 2114: 2113: 2112: 2109: 2106: 2101: 2098: 2095: 2091: 2087: 2083: 2082: 2081: 2077: 2073: 2069: 2065: 2061: 2057: 2053: 2049: 2045: 2044: 2043: 2042: 2039: 2036: 2032: 2024: 2022: 2021: 2018: 2015: 1998: 1971: 1968: 1950: 1947: 1944: 1940: 1939: 1938: 1937: 1934: 1931: 1918: 1915: 1912: 1908: 1904: 1900: 1896: 1890: 1887: 1884: 1880: 1878: 1873: 1870: 1867: 1863: 1860: 1856: 1851: 1848: 1844: 1840: 1839: 1838: 1834: 1830: 1825: 1824: 1823: 1820: 1817: 1812: 1808: 1804: 1803:Line geometry 1800: 1796: 1792: 1788: 1787: 1786: 1782: 1778: 1774: 1770: 1769:Steiner chain 1766: 1765: 1764: 1761: 1758: 1754: 1748: 1743: 1739: 1734: 1730: 1725: 1721: 1716: 1715: 1713: 1695: 1692: 1689: 1686: 1662: 1642: 1639: 1631: 1627: 1609: 1606: 1602: 1593: 1577: 1572: 1566: 1560: 1552: 1549: 1544: 1541: 1536: 1533: 1526: 1523: 1518: 1512: 1504: 1501: 1496: 1493: 1486: 1483: 1478: 1475: 1470: 1464: 1456: 1453: 1446: 1443: 1438: 1435: 1430: 1427: 1422: 1416: 1407: 1402: 1399: 1392: 1391: 1377: 1357: 1354: 1351: 1348: 1345: 1340: 1336: 1327: 1326: 1325: 1321: 1317: 1313: 1309: 1305: 1304: 1303: 1302: 1299: 1296: 1288: 1284: 1280: 1276: 1269: 1264: 1263: 1262: 1256: 1253: 1249: 1245: 1241: 1237: 1232: 1231: 1230: 1223: 1215: 1212: 1209: 1205: 1204: 1203: 1199: 1195: 1190: 1186: 1185: 1184: 1181: 1178: 1173: 1168: 1167: 1166: 1165: 1161: 1157: 1153: 1145: 1131: 1127: 1123: 1119: 1118: 1117: 1113: 1109: 1104: 1103: 1102: 1098: 1094: 1089: 1088: 1087: 1083: 1079: 1074: 1073: 1072: 1068: 1064: 1060: 1059: 1058: 1054: 1050: 1046: 1045: 1044: 1043: 1039: 1035: 1031: 1029: 1022: 1016: 1014: 1013: 1010: 1006: 1000: 997: 991: 989: 988: 984: 980: 976: 959: 958: 954: 950: 946: 937: 933: 929: 925: 921: 920: 919: 917: 913: 909: 908:66.188.89.180 905: 899: 895: 891: 883: 877: 873: 869: 866:mentions. 865: 861: 860: 859: 855: 851: 847: 842: 838: 837: 836: 835: 831: 827: 823: 805: 801: 778: 774: 751: 747: 724: 720: 697: 693: 670: 666: 643: 639: 616: 609: 605: 599: 595: 591: 586: 582: 576: 572: 568: 563: 559: 553: 549: 543: 540: 535: 531: 527: 522: 518: 514: 509: 505: 501: 496: 492: 483: 475: 471: 467: 463: 458: 453: 452: 451: 448: 444: 440: 436: 432: 420: 416: 412: 408: 407:Adam majewski 404: 400: 399: 398: 397: 393: 389: 381: 379: 376: 373: 369: 365: 361: 357: 345: 343: 342: 337: 332: 328: 323: 315: 313: 312: 308: 304: 296: 294: 293: 290: 289:Michael Hardy 286: 278: 264: 261: 257: 250: 249: 244: 241: 240: 238: 234: 233: 228: 224: 220: 217: 213: 212: 196: 192: 186: 183: 182: 179: 162: 158: 154: 153: 145: 139: 134: 132: 129: 125: 124: 120: 114: 111: 108: 104: 99: 95: 89: 81: 72: 71: 61: 56: 55: 48: 44: 40: 36: 35: 34: 28: 25: 22: 18: 17: 4455: 4437: 4422: 4420: 4391: 4382: 4369: 4365: 4361: 4348: 4321: 4306: 4304: 4297: 4289: 4285: 4280: 4277: 4263: 4025: 3901: 3876:Spotchecks: 3863: 3803: 3790: 3777: 3764: 3754: 3744: 3722: 3708: 3694: 3681: 3668: 3658: 3651: 3643:Good Article 3641: 3464: 3388: 3382:EquationNote 3314: 3303: 3293: 3292: 3279: 3268:Instructions 3208: 3118: 2931:— Preceding 2861: 2813:WP:TECHNICAL 2290: 2193: 2028: 1953: 1927: 1855:dual numbers 1737: 1719: 1292: 1278: 1274: 1260: 1239: 1235: 1227: 1188: 1151: 1149: 1027: 1025: 1023: 1020: 1001: 998: 995: 960: 941: 902:— Preceding 897: 889: 887: 481: 479: 424: 385: 377: 349: 326: 321: 319: 316:Organization 300: 282: 246: 243:Did you know 242: 232:Did you know 230: 222: 221:A fact from 191:Low-priority 190: 150: 116:Low‑priority 94:WikiProjects 52: 43:please do so 31: 30: 26: 4058:Thank you @ 3745:broadness ( 3282:transcluded 1986:and vector 1801:or perhaps 1122:Fernandohbc 1093:Fernandohbc 1078:Fernandohbc 975:Vid the Kid 945:Vid the Kid 429:—Preceding 354:—Preceding 237:check views 166:Mathematics 157:mathematics 113:Mathematics 4498:Categories 4454:These are 3683:ref layout 3235:Authorship 3221:GA toolbox 3159:Petrus3743 3141:Petrus3743 2937:Petrus3743 2898:Petrus3743 2864:Petrus3743 2322:is inside 2128:Soddy line 2094:Soddy line 1747:1910.09174 1030:kissing... 476:Existence? 435:SwiftSurge 37:under the 4456:fantastic 4413:Jacobolus 4219:jacobolus 4184:jacobolus 4145:theorem". 4132:jacobolus 4109:technical 4060:jacobolus 4048:jacobolus 4015:jacobolus 3605:jacobolus 3294:Reviewer: 3258:Templates 3249:Reviewing 3192:Passed. — 3185:GA Review 3099:jacobolus 2954:Yes, see 2844:jacobolus 2832:jacobolus 2800:jacobolus 2262:jacobolus 2250:jacobolus 2197:jacobolus 2152:jacobolus 2105:jacobolus 2085:writing.) 2035:jacobolus 2014:jacobolus 1943:jacobolus 1930:jacobolus 1911:jacobolus 1883:jacobolus 1845:, called 1816:jacobolus 1767:FWIW the 1757:jacobolus 1729:0706.0372 1295:jacobolus 1208:jacobolus 1177:jacobolus 841:inversion 331:SlamDiego 303:Daggerbox 297:Curvature 270:Knowledge 227:Main Page 4433:watching 4383:Reviewed 4368:, about 4307:promoted 4239:MOS:LEAD 3931:access). 3653:Criteria 3389:equation 3307:contribs 3263:Criteria 2945:contribs 2933:unsigned 2927:GeoGebra 904:unsigned 884:Kissing? 826:Jowa fan 443:contribs 431:unsigned 368:contribs 360:Skytopia 356:unsigned 84:GA-class 47:reassess 4392:Comment 4329:Comment 4180:MOS:VAR 3889:19 fine 3880:6 fine, 3766:neutral 3755:focus ( 3659:prose ( 2048:WP:GACR 1255:2321856 1024:May be 898:kissing 890:kissing 327:history 229:in the 193:on the 4411:) and 4072:WP:WTW 3779:stable 3695:cites 1907:p. 408 1903:p. 351 1899:p. 109 1370:where 1076:sense. 1063:Jumpow 1034:Jumpow 1009:MaxEnt 482:exists 90:scale. 54:Review 4483:ctrbs 4470:ezlev 4243:Kusma 4197:Kusma 4080:Kusma 4044:Kusma 3973:Inre 3959:Kusma 3905:Kusma 3898:46 ok 3866:Kusma 3725:WP:CV 3711:WP:OR 3697:WP:RS 3618:Kusma 3467:Kusma 3359:any). 3355:case. 3317:Kusma 3297:Kusma 3284:from 3194:Kusma 1742:arXiv 1724:arXiv 1271:(PDF) 1252:JSTOR 1189:least 1152:where 1028:three 4475:user 4444:talk 4417:talk 4409:talk 4362:ALT1 4333:view 4314:talk 4247:talk 4201:talk 4182:). – 4169:talk 4121:talk 4098:talk 4084:talk 4033:talk 3963:talk 3948:talk 3909:talk 3870:talk 3802:6b. 3789:6a. 3753:3b. 3743:3a. 3721:2d. 3707:2c. 3693:2b. 3680:2a. 3667:1b. 3657:1a. 3622:talk 3471:talk 3398:so). 3336:talk 3321:talk 3301:talk 3198:talk 3163:talk 3145:talk 3127:talk 3030:and 2941:talk 2917:talk 2902:talk 2888:talk 2868:talk 2821:talk 2278:talk 2168:talk 2140:talk 2126:and 2076:talk 1833:talk 1781:talk 1320:talk 1198:talk 1160:talk 1156:yoyo 1126:talk 1112:talk 1097:talk 1082:talk 1067:talk 1053:talk 1038:talk 928:talk 912:talk 872:talk 868:yoyo 854:talk 830:talk 766:and 685:and 466:talk 439:talk 411:talk 405:. -- 392:talk 364:talk 322:gist 307:talk 4479:tlk 4419:). 4331:or 4309:by 4292:or 4222:(t) 4187:(t) 4135:(t) 4115:. — 4051:(t) 4018:(t) 3776:5. 3763:4. 3723:no 3709:no 3670:MoS 3608:(t) 3117:" ( 3102:(t) 2847:(t) 2835:(t) 2803:(t) 2784:124 2771:143 2758:423 2745:123 2718:124 2705:143 2692:423 2679:123 2652:124 2639:143 2626:423 2613:123 2586:124 2573:143 2560:423 2547:123 2265:(t) 2253:(t) 2200:(t) 2155:(t) 2108:(t) 2038:(t) 2017:(t) 1946:(t) 1933:(t) 1914:(t) 1886:(t) 1819:(t) 1760:(t) 1314:. — 1298:(t) 1283:doi 1244:doi 1211:(t) 1180:(t) 185:Low 49:it. 4500:: 4463:. 4450:. 4446:) 4385:: 4324:( 4288:, 4249:) 4203:) 4171:) 4157:") 4123:) 4100:) 4086:) 4035:) 3965:) 3950:) 3911:) 3872:) 3624:) 3583:3. 3570:± 3564:− 3526:⋅ 3520:− 3514:⋅ 3508:− 3502:⋅ 3473:) 3433:− 3385:}} 3379:{{ 3338:) 3323:) 3309:) 3200:) 3165:) 3147:) 3129:) 3047:− 3006:− 2966:− 2947:) 2943:• 2929:? 2919:) 2904:) 2890:) 2870:) 2823:) 2780:Δ 2767:Δ 2754:Δ 2741:Δ 2714:Δ 2701:Δ 2688:Δ 2675:Δ 2648:Δ 2635:Δ 2622:Δ 2609:Δ 2582:Δ 2569:Δ 2556:Δ 2543:Δ 2410:△ 2330:△ 2280:) 2170:) 2142:) 2078:) 1969:− 1905:, 1901:, 1835:) 1783:) 1740:, 1722:, 1687:− 1640:− 1607:− 1561:− 1550:− 1542:− 1534:− 1524:− 1513:− 1502:− 1494:− 1484:− 1476:− 1465:− 1454:− 1444:− 1436:− 1428:− 1417:− 1322:) 1279:83 1277:, 1273:, 1250:, 1240:87 1238:, 1200:) 1162:) 1128:) 1114:) 1099:) 1084:) 1069:) 1055:) 1040:) 1032:? 930:) 914:) 874:) 856:) 832:) 541:± 468:) 445:) 441:• 413:) 394:) 370:) 366:• 336:←T 309:) 63:). 4485:) 4481:/ 4477:/ 4473:( 4442:( 4438:— 4423:2 4415:( 4407:( 4343:) 4300:. 4245:( 4233:@ 4199:( 4167:( 4119:( 4096:( 4082:( 4066:@ 4031:( 4027:— 4013:– 4006:⁠ 3992:i 3988:k 3979:⁠ 3961:( 3957:— 3946:( 3942:— 3907:( 3903:— 3868:( 3810:) 3800:) 3794:( 3787:) 3781:( 3774:) 3768:( 3761:) 3751:) 3733:) 3727:( 3719:) 3713:( 3705:) 3699:( 3691:) 3685:( 3678:) 3672:( 3665:) 3655:: 3620:( 3616:— 3580:= 3575:0 3567:1 3561:2 3558:+ 3555:2 3535:0 3532:= 3529:1 3523:2 3517:1 3511:2 3505:2 3499:2 3469:( 3449:0 3446:= 3442:k 3436:1 3428:Q 3420:T 3414:k 3334:( 3319:( 3304:· 3299:( 3196:( 3161:( 3143:( 3125:( 3085:c 3082:, 3079:b 3076:, 3073:a 3053:, 3050:c 3044:b 3041:+ 3038:a 3018:, 3015:c 3012:+ 3009:b 3003:a 2984:, 2981:c 2978:+ 2975:b 2972:+ 2969:a 2939:( 2915:( 2911:— 2900:( 2886:( 2866:( 2819:( 2798:– 2776:= 2763:+ 2750:+ 2710:+ 2697:= 2684:+ 2644:+ 2631:+ 2618:= 2578:+ 2565:+ 2552:= 2520:4 2516:P 2492:4 2488:P 2465:4 2461:P 2438:3 2434:P 2428:2 2424:P 2418:1 2414:P 2388:4 2384:P 2363:, 2358:3 2354:P 2348:2 2344:P 2338:1 2334:P 2308:4 2304:P 2291:@ 2276:( 2234:2 2230:r 2226:+ 2221:1 2217:r 2195:– 2166:( 2138:( 2103:– 2074:( 2029:@ 1999:, 1995:k 1972:1 1964:Q 1881:– 1861:. 1831:( 1827:— 1779:( 1755:– 1744:: 1726:: 1699:] 1696:1 1693:, 1690:1 1684:[ 1663:0 1643:1 1610:1 1603:Q 1578:. 1573:] 1567:1 1553:1 1545:1 1537:1 1527:1 1519:1 1505:1 1497:1 1487:1 1479:1 1471:1 1457:1 1447:1 1439:1 1431:1 1423:1 1408:[ 1403:= 1400:Q 1378:k 1358:, 1355:0 1352:= 1349:k 1346:Q 1341:T 1337:k 1318:( 1293:– 1285:: 1246:: 1196:( 1192:— 1158:( 1124:( 1110:( 1106:— 1095:( 1080:( 1065:( 1051:( 1036:( 983:c 981:/ 979:t 977:( 971:4 967:4 963:4 953:c 951:/ 949:t 947:( 926:( 910:( 870:( 852:( 828:( 806:3 802:k 779:2 775:k 752:1 748:k 725:3 721:k 698:2 694:k 671:1 667:k 644:3 640:k 617:. 610:1 606:k 600:3 596:k 592:+ 587:3 583:k 577:2 573:k 569:+ 564:2 560:k 554:1 550:k 544:2 536:3 532:k 528:+ 523:2 519:k 515:+ 510:1 506:k 502:= 497:4 493:k 464:( 437:( 409:( 390:( 362:( 305:( 262:. 197:. 96::

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index