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119:
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874:
109:
89:
42:
1981:. Actually we can see more than three direct similarities on this first image, because every regular polygon is invariant under certain direct similarities, more precisely certain rotations the center of which is the center of the polygon, and a composition of direct similarities is also a direct similarity. For example we see the image of the initial regular
209:) are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.
99:
3157:
419:
583:
1697:
of similar figures is equal to the square of the ratio of corresponding lengths of those figures (for example, when the side of a square or the radius of a circle is multiplied by three, its area is multiplied by nine — i.e. by three squared). The altitudes of similar triangles are in the same ratio
727:
4179:: adding or subtracting the logarithm of two to the logarithm of one of these numbers produces the logarithm of another of these numbers. In the given set of numbers themselves, this corresponds to a similarity transformation in which the numbers are multiplied or divided by two.
3654:
166:
If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure.
174:
shapes are similar, with a scale factor of 1. However, some school textbooks specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar.
3956:
1840:
of similar figures is equal to the cube of the ratio of corresponding lengths of those figures (for example, when the edge of a cube or the radius of a sphere is multiplied by three, its volume is multiplied by 27 — i.e. by three cubed).
764:
and one other side have lengths in the same ratio. There are several equivalent conditions in this case, such as the right triangles having an acute angle of the same measure, or having the lengths of the legs (sides) being in the same
2700:
278:
889:
and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles (otherwise, for example, all
154:
are not all similar to each other. This is because two ellipses can have different width to height ratios, two rectangles can have different length to breadth ratios, and two isosceles triangles can have different base angles.
439:
3302:
3477:
611:
1831:
1677:
1409:
Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments. Similarities preserve angles but do not necessarily preserve orientation,
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with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence (even if clockwise for one polygon and counterclockwise for the other) are
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1402:
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1833:
Similar figures which can be decomposed into similar triangles will have areas related in the same way. The relationship holds for figures that are not rectifiable as well.
2077:
1301:
1522:
588:
This is known as the SAS similarity criterion. The "SAS" is a mnemonic: each one of the two S's refers to a "side"; the A refers to an "angle" between the two sides.
77:. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is
2632:
830:) the SAS similarity criterion given above was used to replace both Euclid's parallel postulate and the SAS axiom which enabled the dramatic shortening of
4615:
35:
414:{\displaystyle {\frac {\overline {AB}}{\overline {A'B'}}}={\frac {\overline {BC}}{\overline {B'C'}}}={\frac {\overline {AC}}{\overline {A'C'}}}.}
3686:
1761:
4821:
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4697:
4625:
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785:
3163:
3003:
3737:. The similarity is a function such that its value is greater when two points are closer (contrary to the distance, which is a measure of
578:{\displaystyle {\frac {\overline {AB}}{\overline {A'B'}}}={\frac {\overline {BC}}{\overline {B'C'}}},\quad \angle ABC\cong \angle A'B'C'.}
197:
are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of
2116:
1873:
4944:
4902:
4856:
4783:
4741:
4719:
3744:
The definition of the similarity can vary among authors, depending on which properties are desired. The basic common properties are
3708:
2760:
1900:
4868:(1969) . "§5 Similarity in the Euclidean Plane". pp. 67–76. "§7 Isometry and Similarity in Euclidean Space". pp. 96–104.
3649:{\displaystyle \mu ^{D}(f_{s_{1}}\circ f_{s_{2}}\circ \cdots \circ f_{s_{n}}(K))=(r_{s_{1}}\cdot r_{s_{2}}\cdots r_{s_{n}})^{D}.\,}
1844:
Galileo's square–cube law concerns similar solids. If the ratio of similitude (ratio of corresponding sides) between the solids is
3311:
1625:
3389:
898:
would be similar). A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional.
841:(without the use of coordinates) proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the
61:, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly
3754:
722:{\displaystyle {\begin{aligned}\triangle ABC&\sim \triangle A'B'C'\\\triangle ABC&\nsim \triangle A'B'C'\end{aligned}}}
4865:
3690:
1194:
2907:
2781:
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The intuition for the notion of geometric similarity already appears in human children, as can be seen in their drawings.
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202:
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894:
would be similar). Likewise, equality of all angles in sequence is not sufficient to guarantee similarity (otherwise all
4528:
4056:
4949:
1622:
of the large triangle shows that it is similar to the small triangle with an area ratio of 5. The similarity ratio is
31:
4954:
4667:
This traditional term, as explained in its article, is a misnomer. This is actually the 1-dimensional complex line.
3970:
2622:
954:
921:
Several types of curves have the property that all examples of that type are similar to each other. These include:
3679:
2455:
4121:
The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).
2282:
425:
1614:
4435:
2337:
1175:
216:
Any two pairs of angles are congruent, which in
Euclidean geometry implies that all three angles are congruent:
74:
2411:
2017:
1311:
1529:
1040:
4923:
4282:
4277:
3093:
This weaker version applies when the metric is an effective resistance on a topologically self-similar set.
1977:
under the similarity, followed by a red segment going to the following image of vertex, and so on to form a
1939:
1477:. The 2D similarity transformations can then be expressed in terms of complex arithmetic and are given by
1372:
4655:
4247:
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2005:
1717:
846:
842:
70:
66:
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4231:
4125:
3372:
1463:
929:
823:
171:
78:
3951:{\displaystyle S(a,b)\leq S(a,a)\quad {\text{and}}\quad \forall (a,b),S(a,b)=S(a,a)\Leftrightarrow a=b}
854:
4287:
212:
There are several criteria each of which is necessary and sufficient for two triangles to be similar:
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4369:
4337:
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2001:
1911:
1881:
1179:
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978:
827:
736:
139:
1688:
4542:
4272:
3451:
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1974:
1422:
858:
831:
816:
750:
743:
433:
Any two pairs of sides are proportional, and the angles included between these sides are congruent:
198:
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2009:
1954:
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984:
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151:
62:
50:
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2744:
1958:
1957:, the vertices of which are each on a side of the previous polygon. This rotational reduction
1364:
850:
4707:
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4617:
Understanding
Similarity: Bridging Geometric and Numeric Contexts for Proportional Reasoning
4462:
4241:
2086:
2082:
118:
4152:
2695:{\displaystyle {\overset {}{{\overrightarrow {PE}},{\overrightarrow {PA}}=+135^{\circ }}}}
2618:
1950:
1891:
1680:
1441:
1433:
1009:
925:
906:
158:
731:
There are several elementary results concerning similar triangles in
Euclidean geometry:
4753:
757:
4191:
4933:
4257:
2725:
1962:
1949:
On the first image below the title, on the left, one or another similarity shrinks a
1470:
4635:
2097:
of the three transformations: rotation, homothety and similarity. For example point
4918:
4729:
3726:
2854:
2277:
we can also compose in any order a rotation of –45° angle and a homothety of ratio
1970:
1619:
1459:
935:
862:
4492:
3668:
1474:
1020:
873:
808:
17:
1942:: a point that the similarity keeps unchanged, then this only point is called "
2771:
1019:
from the space onto itself that multiplies all distances by the same positive
761:
2008:
and a homothety. Similarity and rotation have the same angle of +135 degrees
1986:
1013:
971:
950:
895:
742:
Two triangles, both similar to a third triangle, are similar to each other (
147:
108:
424:
This is equivalent to saying that one triangle (or its mirror image) is an
4847:
88:
4296:
4124:
Note that, in the topological sense used here, a similarity is a kind of
3734:
3722:
1982:
1969:
of the similarity is the common center of the successive polygons. A red
965:
945:
592:
Symbolically, we write the similarity and dissimilarity of two triangles
41:
3693: in this section. Unsourced material may be challenged and removed.
959:
891:
882:
143:
27:
Property of objects which are scaled or mirrored versions of each other
3297:{\displaystyle {\begin{aligned}z'&=0.1\\z'&=0.1\end{aligned}}}
98:
2273:, the last decomposition being only represented on the image. To get
1993:, which is a similarity of ±180° angle and a positive ratio equal to
1978:
1837:
940:
819:(where Wallis's postulate is false) similar triangles are congruent.
135:
131:
753:
of similar triangles have the same ratio as the corresponding sides.
4893:
Martin, George E. (1982). "Chapter 13: Similarities in the Plane".
3472:
are "small", we have the following simple formula for the measure:
1454:
of isometries also forms a normal subgroup. The similarities group
4543:
The shape of an ellipse or hyperbola depends only on the ratio b/a
3155:
2000:
Below the title on the right, the second image shows a similarity
1613:
872:
117:
107:
97:
87:
58:
40:
2247:
can be decomposed into a rotation and a homothety of same center
3156:
1917:
1694:
2883:
into itself that multiplies all distances by the same positive
4690:
Experiencing
Geometry/Euclidean and Non-Euclidean with History
4186:
3662:
2732:. This set of points is the blue quarter of circle of center
205:. It can be shown that two triangles having congruent angles (
1672:{\displaystyle {\tfrac {5}{h}}={\tfrac {h}{1}}={\sqrt {5}}.}
81:
to the result of a particular uniform scaling of the other.
4620:(Ph.D.). Kalamazoo, Michigan: Western Michigan University.
1698:
as corresponding sides. If a triangle has a side of length
1706:
then a similar triangle with corresponding side of length
822:
In the axiomatic treatment of
Euclidean geometry given by
1826:{\displaystyle A'={\frac {1}{2}}\cdot kb\cdot kh=k^{2}A.}
4171:
ranges over all integers. When this set is plotted on a
2539:
center of a rotation of +135° angle that transforms ray
1848:, then the ratio of surface areas of the solids will be
4413:
4411:
4203:
2226:
the previous rotation, homothety and similarity, with “
1237:{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{n},}
4471:. American Mathematical Society. Theorem 120, p. 125.
3160:
A self-similar set constructed with two similitudes:
3084:{\displaystyle \lim {\frac {d(f(x),f(y))}{d(x,y)}}=r.}
2986:
Weaker versions of similarity would for instance have
2786:
2469:
2416:
2287:
2022:
1728:
1645:
1630:
4059:
3973:
3829:
3757:
3480:
3392:
3314:
3166:
3006:
2910:
2838:{\displaystyle {\tfrac {\log 3}{\log 2}}=\log _{2}3,}
2784:
2635:
2586:
2547:
2458:
2414:
2285:
2119:
2061:
2020:
1764:
1720:
1628:
1532:
1486:
1375:
1314:
1259:
1197:
1186:
if one is the image of the other under a similarity.
1043:
861:. Similar triangles also provide the foundations for
614:
442:
281:
4895:
Transformation
Geometry: An Introduction to Symmetry
3816:
Majored by the similarity of one element on itself (
3741:: the closer the points, the lesser the distance).
3450:which is often (but not always) equal to the set's
3112:for which there exists a finite set of similitudes
2608:{\displaystyle {\overset {}{\overrightarrow {SA}}}}
2569:{\displaystyle {\overset {}{\overrightarrow {SE}}}}
2448:is the center of this similarity because any point
1710:will have an altitude drawn to that side of length
4752:
4110:
4039:
3950:
3805:
3648:
3440:
3361:
3296:
3083:
2976:
2894:'s contraction factor, so that for any two points
2837:
2694:
2607:
2568:
2492:
2434:
2312:
2206:
2071:
2040:
2012:. Similarity and homothety have the same ratio of
1825:
1750:
1671:
1569:
1516:
1396:
1344:
1295:
1236:
1110:
721:
577:
413:
65:(enlarging or reducing), possibly with additional
4161:{..., 0.5, 0.75, 1, 1.5, 2, 3, 4, 6, 8, 12, ...}
4111:{\displaystyle \forall (a,b)\ S(a,b)<\infty .}
2207:{\displaystyle T=H(W)=(R(F))=(H\circ R)(F)=D(F),}
1927:Examples of direct similarities that have each a
4658:where the triangle angle sum is not 180 degrees.
4357:
3007:
2233:This direct similarity that transforms triangle
1758:while the area of the similar triangle will be
1155:has many names in the literature including; the
4755:The Geometric Viewpoint/A Survey of Geometries
1425:under the operation of composition called the
142:are similar to each other. On the other hand,
4141:
4040:{\displaystyle \forall (a,b)\ S(a,b)=S(b,a),}
1973:joins a vertex of the initial polygon to its
1961:, so the initial polygon is extended into an
1702:and an altitude drawn to that side of length
1473:, that is, as a 2-dimensional space over the
837:Similar triangles provide the basis for many
272:All the corresponding sides are proportional:
8:
3371:These self-similar sets have a self-similar
3362:{\displaystyle \bigcup _{s\in S}f_{s}(K)=K.}
2493:{\displaystyle AK={\tfrac {AK}{\sqrt {2}}},}
4919:Animated demonstration of similar triangles
4468:Lessons in Geometry, Vol. I: Plane Geometry
3441:{\displaystyle \sum _{s\in S}(r_{s})^{D}=1}
2778:. A space having self-similarity dimension
2313:{\displaystyle {\tfrac {-{\sqrt {2}}}{2}}.}
1421:The similarities of Euclidean space form a
162:Figures shown in the same color are similar
3806:{\displaystyle \forall (a,b),S(a,b)\geq 0}
36:Similarity transformation (disambiguation)
4926:- an illustrative dynamic geometry sketch
4299:(shell of concentric, similar ellipsoids)
4058:
3972:
3960:More properties can be invoked, such as:
3870:
3828:
3756:
3709:Learn how and when to remove this message
3645:
3636:
3624:
3619:
3604:
3599:
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3544:
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2019:
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1205:
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1086:
1042:
615:
613:
486:
443:
441:
368:
325:
282:
280:
3096:A self-similar subset of a metric space
2770:
2763:point of these two quarters of circles.
2747:of the blue quarter of circle of center
2435:{\displaystyle {\tfrac {1}{\sqrt {2}}}.}
2041:{\displaystyle {\tfrac {\sqrt {2}}{2}},}
1469:One can view the Euclidean plane as the
1345:{\displaystyle A\in O^{n}(\mathbb {R} )}
811:and is logically equivalent to Euclid's
157:
4692:(3rd ed.). Pearson Prentice-Hall.
4647:
4538:
4536:
4314:
4137:
1906:Example of direct similarity of center
1570:{\displaystyle f(z)=a{\overline {z}}+b}
1111:{\displaystyle d(f(x),f(y))=r\,d(x,y),}
150:are not all similar to each other, and
4516:
4504:
4449:
4417:
4353:
4336:, p. 127. This is also proved in
4321:
2716:, of which the two radius leading to
2521:
1884:with itself several times successively
4601:
4577:
4553:
4333:
2977:{\displaystyle d(f(x),f(y))=rd(x,y).}
1854:, while the ratio of volumes will be
1606:, these similarities are isometries.
1397:{\displaystyle t\in \mathbb {R} ^{n}}
881:The concept of similarity extends to
7:
4589:
4565:
3691:adding citations to reliable sources
1751:{\displaystyle A={\tfrac {1}{2}}bh,}
1714:. The area of the first triangle is
807:satisfying this condition exists is
4924:A Fundamental Theorem of Similarity
2845:which is approximately 1.58. (From
2362:similarity that transforms segment
146:are not all similar to each other,
138:are similar to each other, and all
4102:
4060:
3974:
3876:
3758:
2452:being invariant under it fulfills
2113:under the homothety, more briefly
688:
669:
638:
619:
545:
530:
25:
3729:can be constructed by defining a
2251:in several manners. For example,
83:
4190:
3667:
3147:is the unique compact subset of
1938:If a similarity has exactly one
1899:
1872:
1679:This can be used to construct a
1432:. The direct similitudes form a
4734:Geometry/A Comprehensive Course
4439:. Prometheus Books. p. 22.
4244:(string or sequence similarity)
4142:§ In general metric spaces
3875:
3869:
3678:needs additional citations for
529:
134:are similar to each other, all
4897:. Springer. pp. 136–146.
4851:. Springer. pp. 183–189.
4654:This statement is not true in
4096:
4084:
4075:
4063:
4031:
4019:
4010:
3998:
3989:
3977:
3936:
3933:
3921:
3912:
3900:
3891:
3879:
3866:
3854:
3845:
3833:
3794:
3782:
3773:
3761:
3633:
3572:
3566:
3563:
3557:
3491:
3458:. If the overlaps between the
3423:
3409:
3347:
3341:
3287:
3259:
3244:
3241:
3216:
3204:
3192:
3189:
3066:
3054:
3046:
3043:
3037:
3028:
3022:
3016:
2968:
2956:
2944:
2941:
2935:
2926:
2920:
2914:
2730:2(180° – 135°) = 2 × 45° = 90°
2198:
2192:
2183:
2177:
2174:
2162:
2156:
2153:
2147:
2141:
2135:
2129:
2105:under the rotation, and point
1542:
1536:
1496:
1490:
1339:
1331:
1269:
1263:
1216:
1102:
1090:
1077:
1074:
1068:
1059:
1053:
1047:
265:and the triangles are similar.
1:
4849:A Course in Modern Geometries
4795:Geometry/From Euclid to Knots
4778:(5th ed.). Brooks/Cole.
4268:Basic proportionality theorem
1914:into a rotation of 135° angle
1879:Example where each similarity
1026:, so that for any two points
4614:Cox, Dana Christine (2008).
4358:Henderson & Taimiņa 2005
2740:. In the same manner, point
1916:and a homothety that halves
1681:non-periodic infinite tiling
1556:
1462:, so every similarity is an
1458:is itself a subgroup of the
974:function for different bases
746:of similarity of triangles).
520:
497:
477:
454:
402:
379:
359:
336:
316:
293:
2621:problem plus a question of
2072:{\displaystyle {\sqrt {2}}}
1610:Area ratio and volume ratio
1296:{\displaystyle f(x)=rAx+t,}
863:right triangle trigonometry
803:. The statement that point
32:Similarity (disambiguation)
4971:
4812:Venema, Gerard A. (2006).
4751:Sibley, Thomas Q. (1998).
4433:; Lehmann, Ingmar (2012).
4144:sections of this article.
1686:
1174:a similarity is called an
962:of a specific eccentricity
85:
29:
4945:Equivalence (mathematics)
4882:Geometry: An Introduction
4816:. Pearson Prentice-Hall.
4568:, p. 47 Theorem 2.1.
4529:a proof from academia.edu
4405:, Book VI, Proposition 6.
4390:, Book VI, Proposition 5.
4375:, Book VI, Proposition 4.
4343:, Book VI, Proposition 4.
4159:to itself, e.g., the set
4138:§ In Euclidean space
4134:similarity transformation
3132:with contraction factors
1965:of regular polygons. The
1524:(direct similitudes), and
1517:{\displaystyle f(z)=az+b}
1414:preserve orientation and
1406:is a translation vector.
1002:similarity transformation
4870:Introduction to Geometry
4774:Smart, James R. (1998).
4714:. W. H. Freeman and Co.
4436:The Secrets of Triangles
4155:means that a pattern is
2994:function and the scalar
2767:In general metric spaces
2378:, but transforms point
1864:Similarity with a center
928:(any two lines are even
4814:Foundations of Geometry
4283:Similarity (philosophy)
4175:it has one-dimensional
4163:of numbers of the form
1577:(opposite similitudes),
1182:). Two sets are called
258:is equal in measure to
251:then this implies that
244:is equal in measure to
230:is equal in measure to
4880:Ewald, Günter (1971).
4831:Yale, Paul B. (1968).
4656:non-Euclidean geometry
4431:Posamentier, Alfred S.
4248:Helmert transformation
4177:translational symmetry
4112:
4041:
3952:
3807:
3650:
3442:
3363:
3304:
3298:
3085:
2978:
2879:from the metric space
2850:
2839:
2696:
2609:
2570:
2494:
2436:
2332:" like "Indirect", if
2314:
2208:
2073:
2051:multiplicative inverse
2042:
1836:The ratio between the
1827:
1752:
1693:The ratio between the
1684:
1673:
1571:
1518:
1398:
1346:
1297:
1246:a similarity of ratio
1238:
1165:similarity coefficient
1112:
878:
869:Other similar polygons
847:geometric mean theorem
843:angle bisector theorem
723:
579:
415:
163:
123:
113:
103:
93:
57:if they have the same
46:
4874:John Wiley & Sons
4833:Geometry and Symmetry
4303:Solution of triangles
4232:Congruence (geometry)
4157:non-trivially similar
4113:
4042:
3953:
3808:
3651:
3443:
3384:given by the formula
3364:
3299:
3159:
3086:
2979:
2840:
2774:
2697:
2610:
2571:
2527:of direct similarity
2522:construct the center
2495:
2437:
2340:with respect to line
2328:" like "Mirror" and "
2315:
2209:
2074:
2043:
1946:" of the similarity.
1828:
1753:
1674:
1617:
1589:are complex numbers,
1572:
1519:
1464:affine transformation
1399:
1347:
1298:
1239:
1113:
876:
824:George David Birkhoff
737:equilateral triangles
724:
580:
416:
207:equiangular triangles
161:
140:equilateral triangles
121:
111:
101:
91:
44:
4888:. pp. 106, 181.
4886:Wadsworth Publishing
4793:Stahl, Saul (2003).
4057:
3971:
3827:
3755:
3687:improve this article
3478:
3390:
3312:
3164:
3004:
2908:
2782:
2702:is an arc of circle
2633:
2625:. The set of points
2584:
2545:
2535:, how to find point
2507:, otherwise written
2456:
2412:
2283:
2117:
2059:
2018:
1762:
1718:
1626:
1530:
1484:
1416:opposite similitudes
1373:
1312:
1257:
1195:
1180:rigid transformation
1041:
979:exponential function
612:
440:
279:
30:For other uses, see
4682:Henderson, David W.
4580:, pp. 179–181.
4452:, pp. 384–393.
4273:Semantic similarity
3452:Hausdorff dimension
2847:Hausdorff dimension
2776:Sierpiński triangle
1890:at the center of a
1157:ratio of similarity
985:Logarithmic spirals
981:for different bases
859:Pythagorean theorem
817:hyperbolic geometry
776:and a line segment
760:are similar if the
199:corresponding sides
152:isosceles triangles
4950:Euclidean geometry
4759:. Addison-Wesley.
4356:, p. 122 and
4292:numerical taxonomy
4253:Inversive geometry
4202:. You can help by
4108:
4037:
3948:
3803:
3646:
3438:
3408:
3359:
3330:
3305:
3294:
3292:
3081:
2974:
2851:
2835:
2811:
2692:
2605:
2566:
2490:
2485:
2432:
2427:
2310:
2305:
2204:
2089:similarity. Point
2069:
2038:
2033:
2010:modulo 360 degrees
1989:of negative ratio
1823:
1748:
1737:
1685:
1669:
1654:
1639:
1567:
1514:
1427:similarities group
1412:direct similitudes
1394:
1342:
1293:
1234:
1138:Euclidean distance
1108:
992:In Euclidean space
879:
877:Similar rectangles
855:Menelaus's theorem
813:parallel postulate
809:Wallis's postulate
719:
717:
575:
411:
164:
124:
114:
104:
94:
53:, two objects are
51:Euclidean geometry
47:
4955:Triangle geometry
4866:Coxeter, H. S. M.
4823:978-0-13-143700-5
4804:978-0-13-032927-1
4797:. Prentice-Hall.
4776:Modern Geometries
4766:978-0-201-87450-1
4708:Jacobs, Harold R.
4699:978-0-13-143748-7
4627:978-0-549-75657-6
4478:978-0-8218-4367-3
4463:Hadamard, Jacques
4278:Similarity search
4237:Spiral similarity
4220:
4219:
4173:logarithmic scale
4080:
3994:
3873:
3748:Positive defined:
3719:
3718:
3711:
3456:packing dimension
3393:
3315:
3070:
2810:
2690:
2689:
2670:
2652:
2603:
2602:
2601:
2564:
2563:
2562:
2500:only possible if
2484:
2483:
2426:
2425:
2402:under similarity
2304:
2298:
2230:" like "Direct".
2067:
2032:
2028:
1784:
1736:
1664:
1653:
1638:
1559:
1365:orthogonal matrix
1161:stretching factor
828:Birkhoff's axioms
786:ruler and compass
769:Given a triangle
524:
523:
500:
481:
480:
457:
406:
405:
382:
363:
362:
339:
320:
319:
296:
179:Similar triangles
130:For example, all
128:
127:
16:(Redirected from
4962:
4908:
4889:
4862:
4836:
4827:
4808:
4789:
4770:
4758:
4747:
4725:
4703:
4668:
4665:
4659:
4652:
4640:
4639:
4611:
4605:
4599:
4593:
4587:
4581:
4575:
4569:
4563:
4557:
4551:
4545:
4540:
4531:
4526:
4520:
4514:
4508:
4502:
4496:
4489:
4483:
4482:
4459:
4453:
4447:
4441:
4440:
4427:
4421:
4415:
4406:
4397:
4391:
4382:
4376:
4367:
4361:
4350:
4344:
4331:
4325:
4319:
4288:Similarity space
4242:Hamming distance
4215:
4212:
4194:
4187:
4170:
4166:
4162:
4132:the same as the
4128:. This usage is
4117:
4115:
4114:
4109:
4078:
4046:
4044:
4043:
4038:
3992:
3957:
3955:
3954:
3949:
3874:
3871:
3812:
3810:
3809:
3804:
3714:
3707:
3703:
3700:
3694:
3671:
3663:
3655:
3653:
3652:
3647:
3641:
3640:
3631:
3630:
3629:
3628:
3611:
3610:
3609:
3608:
3591:
3590:
3589:
3588:
3556:
3555:
3554:
3553:
3530:
3529:
3528:
3527:
3510:
3509:
3508:
3507:
3490:
3489:
3471:
3447:
3445:
3444:
3439:
3431:
3430:
3421:
3420:
3407:
3383:
3379:
3368:
3366:
3365:
3360:
3340:
3339:
3329:
3303:
3301:
3300:
3295:
3293:
3271:
3270:
3230:
3178:
3150:
3146:
3142:
3131:
3111:
3107:
3090:
3088:
3087:
3082:
3071:
3069:
3049:
3011:
2997:
2989:
2983:
2981:
2980:
2975:
2901:
2897:
2893:
2889:
2882:
2878:
2867:
2844:
2842:
2841:
2836:
2825:
2824:
2812:
2809:
2798:
2787:
2758:
2754:
2750:
2743:
2739:
2735:
2731:
2723:
2719:
2715:
2711:
2707:
2706:
2701:
2699:
2698:
2693:
2691:
2688:
2687:
2686:
2671:
2666:
2658:
2653:
2648:
2640:
2637:
2628:
2616:
2614:
2612:
2611:
2606:
2604:
2597:
2589:
2588:
2577:
2575:
2573:
2572:
2567:
2565:
2558:
2550:
2549:
2538:
2534:
2530:
2525:
2516:
2506:
2499:
2497:
2496:
2491:
2486:
2479:
2478:
2470:
2451:
2447:
2443:
2441:
2439:
2438:
2433:
2428:
2421:
2417:
2405:
2401:
2398:is the image of
2397:
2393:
2389:
2385:
2381:
2377:
2376:
2371:
2367:
2366:
2357:
2343:
2335:
2331:
2327:
2321:
2319:
2317:
2316:
2311:
2306:
2300:
2299:
2294:
2288:
2276:
2272:
2250:
2246:
2239:
2229:
2225:
2221:
2217:
2213:
2211:
2210:
2205:
2112:
2109:is the image of
2108:
2104:
2101:is the image of
2100:
2092:
2083:square root of 2
2080:
2078:
2076:
2075:
2070:
2068:
2063:
2049:
2047:
2045:
2044:
2039:
2034:
2024:
2023:
1996:
1992:
1932:
1909:
1903:
1894:that it shrinks.
1876:
1859:
1853:
1847:
1832:
1830:
1829:
1824:
1816:
1815:
1785:
1777:
1772:
1757:
1755:
1754:
1749:
1738:
1729:
1713:
1709:
1705:
1701:
1678:
1676:
1675:
1670:
1665:
1660:
1655:
1646:
1640:
1631:
1605:
1603:
1595:
1588:
1584:
1576:
1574:
1573:
1568:
1560:
1552:
1523:
1521:
1520:
1515:
1457:
1453:
1439:
1431:
1405:
1403:
1401:
1400:
1395:
1393:
1392:
1387:
1363:
1353:
1351:
1349:
1348:
1343:
1338:
1330:
1329:
1302:
1300:
1299:
1294:
1249:
1245:
1243:
1241:
1240:
1235:
1230:
1229:
1224:
1215:
1214:
1209:
1173:
1154:
1147:
1143:
1135:
1117:
1115:
1114:
1109:
1033:
1029:
1025:
1018:
987:are self-similar
910:
904:
832:Hilbert's axioms
806:
802:
791:
784:one can, with a
783:
782:
775:
728:
726:
725:
720:
718:
714:
706:
698:
664:
656:
648:
605:
598:
584:
582:
581:
576:
571:
563:
555:
525:
519:
518:
510:
501:
496:
488:
487:
482:
476:
475:
467:
458:
453:
445:
444:
420:
418:
417:
412:
407:
401:
400:
392:
383:
378:
370:
369:
364:
358:
357:
349:
340:
335:
327:
326:
321:
315:
314:
306:
297:
292:
284:
283:
264:
257:
250:
243:
236:
229:
196:
189:
84:
21:
18:Similar triangle
4970:
4969:
4965:
4964:
4963:
4961:
4960:
4959:
4930:
4929:
4915:
4905:
4892:
4879:
4859:
4846:
4843:
4841:Further reading
4830:
4824:
4811:
4805:
4792:
4786:
4773:
4767:
4750:
4744:
4728:
4722:
4706:
4700:
4680:
4677:
4672:
4671:
4666:
4662:
4653:
4649:
4644:
4643:
4628:
4613:
4612:
4608:
4600:
4596:
4588:
4584:
4576:
4572:
4564:
4560:
4552:
4548:
4541:
4534:
4527:
4523:
4515:
4511:
4503:
4499:
4490:
4486:
4479:
4461:
4460:
4456:
4448:
4444:
4429:
4428:
4424:
4416:
4409:
4398:
4394:
4383:
4379:
4368:
4364:
4351:
4347:
4332:
4328:
4320:
4316:
4311:
4263:Proportionality
4228:
4216:
4210:
4207:
4200:needs expansion
4185:
4168:
4164:
4160:
4153:Self-similarity
4150:
4148:Self-similarity
4055:
4054:
3969:
3968:
3825:
3824:
3818:auto-similarity
3753:
3752:
3715:
3704:
3698:
3695:
3684:
3672:
3661:
3632:
3620:
3615:
3600:
3595:
3580:
3575:
3545:
3540:
3519:
3514:
3499:
3494:
3481:
3476:
3475:
3464:
3459:
3422:
3412:
3388:
3387:
3381:
3380:with dimension
3375:
3331:
3310:
3309:
3291:
3290:
3262:
3231:
3223:
3220:
3219:
3179:
3171:
3162:
3161:
3148:
3144:
3139:
3133:
3130:
3119:
3113:
3109:
3097:
3050:
3012:
3002:
3001:
2995:
2987:
2906:
2905:
2899:
2895:
2891:
2887:
2880:
2876:
2857:
2816:
2799:
2788:
2780:
2779:
2769:
2756:
2752:
2748:
2741:
2737:
2733:
2729:
2721:
2717:
2713:
2709:
2704:
2703:
2678:
2659:
2641:
2638:
2631:
2630:
2626:
2619:inscribed angle
2590:
2582:
2581:
2579:
2551:
2543:
2542:
2540:
2536:
2532:
2528:
2523:
2508:
2501:
2471:
2454:
2453:
2449:
2445:
2410:
2409:
2407:
2403:
2399:
2395:
2394:itself. Square
2391:
2387:
2383:
2379:
2374:
2373:
2369:
2364:
2363:
2345:
2341:
2333:
2329:
2325:
2289:
2281:
2280:
2278:
2274:
2252:
2248:
2241:
2234:
2227:
2223:
2219:
2215:
2115:
2114:
2110:
2106:
2102:
2098:
2090:
2057:
2056:
2054:
2016:
2015:
2013:
1994:
1990:
1951:regular polygon
1940:invariant point
1936:
1935:
1934:
1933:
1926:
1923:
1922:
1921:
1915:
1910:
1907:
1904:
1896:
1895:
1892:regular polygon
1885:
1880:
1877:
1866:
1855:
1849:
1845:
1807:
1765:
1760:
1759:
1716:
1715:
1711:
1707:
1703:
1699:
1691:
1689:Square–cube law
1624:
1623:
1612:
1599:
1597:
1590:
1586:
1582:
1528:
1527:
1482:
1481:
1455:
1444:
1442:Euclidean group
1437:
1434:normal subgroup
1429:
1382:
1371:
1370:
1368:
1355:
1321:
1310:
1309:
1307:
1255:
1254:
1250:takes the form
1247:
1219:
1204:
1193:
1192:
1190:
1168:
1152:
1145:
1141:
1122:
1039:
1038:
1031:
1027:
1023:
1016:
1010:Euclidean space
1000:(also called a
994:
919:
908:
902:
871:
804:
793:
789:
788:, find a point
778:
777:
770:
758:right triangles
716:
715:
707:
699:
691:
681:
666:
665:
657:
649:
641:
631:
610:
609:
600:
593:
564:
556:
548:
511:
503:
502:
489:
468:
460:
459:
446:
438:
437:
393:
385:
384:
371:
350:
342:
341:
328:
307:
299:
298:
285:
277:
276:
259:
252:
245:
238:
231:
224:
191:
184:
183:Two triangles,
181:
45:Similar figures
39:
28:
23:
22:
15:
12:
11:
5:
4968:
4966:
4958:
4957:
4952:
4947:
4942:
4932:
4931:
4928:
4927:
4921:
4914:
4913:External links
4911:
4910:
4909:
4903:
4890:
4877:
4863:
4857:
4842:
4839:
4838:
4837:
4828:
4822:
4809:
4803:
4790:
4784:
4771:
4765:
4748:
4742:
4726:
4720:
4704:
4698:
4686:Taimiņa, Daina
4676:
4673:
4670:
4669:
4660:
4646:
4645:
4642:
4641:
4626:
4606:
4604:, p. 182.
4594:
4582:
4570:
4558:
4546:
4532:
4521:
4519:, p. 145.
4509:
4507:, p. 122.
4497:
4484:
4477:
4454:
4442:
4422:
4420:, p. 143.
4407:
4392:
4377:
4362:
4360:, p. 123.
4352:For instance,
4345:
4326:
4313:
4312:
4310:
4307:
4306:
4305:
4300:
4294:
4285:
4280:
4275:
4270:
4265:
4260:
4255:
4250:
4245:
4239:
4234:
4227:
4224:
4218:
4217:
4197:
4195:
4184:
4181:
4149:
4146:
4119:
4118:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4083:
4077:
4074:
4071:
4068:
4065:
4062:
4048:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3991:
3988:
3985:
3982:
3979:
3976:
3947:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3822:
3821:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3760:
3750:
3749:
3717:
3716:
3675:
3673:
3666:
3660:
3657:
3644:
3639:
3635:
3627:
3623:
3618:
3614:
3607:
3603:
3598:
3594:
3587:
3583:
3578:
3574:
3571:
3568:
3565:
3562:
3559:
3552:
3548:
3543:
3539:
3536:
3533:
3526:
3522:
3517:
3513:
3506:
3502:
3497:
3493:
3488:
3484:
3462:
3437:
3434:
3429:
3425:
3419:
3415:
3411:
3406:
3403:
3400:
3396:
3358:
3355:
3352:
3349:
3346:
3343:
3338:
3334:
3328:
3325:
3322:
3318:
3307:
3306:
3289:
3286:
3283:
3280:
3277:
3274:
3269:
3265:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3232:
3229:
3226:
3222:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3180:
3177:
3174:
3170:
3169:
3137:
3122:
3117:
3080:
3077:
3074:
3068:
3065:
3062:
3059:
3056:
3053:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3009:
2973:
2970:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2834:
2831:
2828:
2823:
2819:
2815:
2808:
2805:
2802:
2797:
2794:
2791:
2768:
2765:
2751:inside square
2736:inside square
2685:
2681:
2677:
2674:
2669:
2665:
2662:
2656:
2651:
2647:
2644:
2600:
2596:
2593:
2561:
2557:
2554:
2489:
2482:
2477:
2474:
2467:
2464:
2461:
2431:
2424:
2420:
2309:
2303:
2297:
2292:
2240:into triangle
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2093:is the common
2066:
2037:
2031:
2027:
1955:concentric one
1925:
1924:
1905:
1898:
1897:
1878:
1871:
1870:
1869:
1868:
1867:
1865:
1862:
1822:
1819:
1814:
1810:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1783:
1780:
1775:
1771:
1768:
1747:
1744:
1741:
1735:
1732:
1726:
1723:
1687:Main article:
1668:
1663:
1658:
1652:
1649:
1643:
1637:
1634:
1611:
1608:
1579:
1578:
1566:
1563:
1558:
1555:
1550:
1547:
1544:
1541:
1538:
1535:
1525:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1391:
1386:
1381:
1378:
1341:
1337:
1333:
1328:
1324:
1320:
1317:
1304:
1303:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1233:
1228:
1223:
1218:
1213:
1208:
1203:
1200:
1119:
1118:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
993:
990:
989:
988:
982:
977:Graphs of the
975:
970:Graphs of the
968:
963:
957:
953:of a specific
948:
943:
938:
933:
918:
917:Similar curves
915:
870:
867:
851:Ceva's theorem
767:
766:
754:
749:Corresponding
747:
740:
713:
710:
705:
702:
697:
694:
690:
687:
684:
682:
680:
677:
674:
671:
668:
667:
663:
660:
655:
652:
647:
644:
640:
637:
634:
632:
630:
627:
624:
621:
618:
617:
590:
589:
574:
570:
567:
562:
559:
554:
551:
547:
544:
541:
538:
535:
532:
528:
522:
517:
514:
509:
506:
499:
495:
492:
485:
479:
474:
471:
466:
463:
456:
452:
449:
435:
434:
430:
429:
410:
404:
399:
396:
391:
388:
381:
377:
374:
367:
361:
356:
353:
348:
345:
338:
334:
331:
324:
318:
313:
310:
305:
302:
295:
291:
288:
274:
273:
269:
268:
267:
266:
218:
217:
180:
177:
126:
125:
115:
105:
95:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4967:
4956:
4953:
4951:
4948:
4946:
4943:
4941:
4938:
4937:
4935:
4925:
4922:
4920:
4917:
4916:
4912:
4906:
4904:0-387-90636-3
4900:
4896:
4891:
4887:
4883:
4878:
4875:
4871:
4867:
4864:
4860:
4858:0-387-98972-2
4854:
4850:
4845:
4844:
4840:
4835:. Holden-Day.
4834:
4829:
4825:
4819:
4815:
4810:
4806:
4800:
4796:
4791:
4787:
4785:0-534-35188-3
4781:
4777:
4772:
4768:
4762:
4757:
4756:
4749:
4745:
4743:0-486-65812-0
4739:
4735:
4731:
4727:
4723:
4721:0-7167-0456-0
4717:
4713:
4709:
4705:
4701:
4695:
4691:
4687:
4683:
4679:
4678:
4674:
4664:
4661:
4657:
4651:
4648:
4637:
4633:
4629:
4623:
4619:
4618:
4610:
4607:
4603:
4598:
4595:
4592:, p. 46.
4591:
4586:
4583:
4579:
4574:
4571:
4567:
4562:
4559:
4556:, p. 92.
4555:
4550:
4547:
4544:
4539:
4537:
4533:
4530:
4525:
4522:
4518:
4513:
4510:
4506:
4501:
4498:
4494:
4488:
4485:
4480:
4474:
4470:
4469:
4464:
4458:
4455:
4451:
4446:
4443:
4438:
4437:
4432:
4426:
4423:
4419:
4414:
4412:
4408:
4404:
4403:
4396:
4393:
4389:
4388:
4381:
4378:
4374:
4373:
4366:
4363:
4359:
4355:
4349:
4346:
4342:
4341:
4335:
4330:
4327:
4324:, p. 35.
4323:
4318:
4315:
4308:
4304:
4301:
4298:
4295:
4293:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4258:Jaccard index
4256:
4254:
4251:
4249:
4246:
4243:
4240:
4238:
4235:
4233:
4230:
4229:
4225:
4223:
4214:
4205:
4201:
4198:This section
4196:
4193:
4189:
4188:
4182:
4180:
4178:
4174:
4158:
4154:
4147:
4145:
4143:
4139:
4135:
4131:
4127:
4122:
4105:
4099:
4093:
4090:
4087:
4081:
4072:
4069:
4066:
4052:
4049:
4034:
4028:
4025:
4022:
4016:
4013:
4007:
4004:
4001:
3995:
3986:
3983:
3980:
3966:
3963:
3962:
3961:
3958:
3945:
3942:
3939:
3930:
3927:
3924:
3918:
3915:
3909:
3906:
3903:
3897:
3894:
3888:
3885:
3882:
3863:
3860:
3857:
3851:
3848:
3842:
3839:
3836:
3830:
3819:
3815:
3814:
3813:
3800:
3797:
3791:
3788:
3785:
3779:
3776:
3770:
3767:
3764:
3747:
3746:
3745:
3742:
3740:
3739:dissimilarity
3736:
3733:instead of a
3732:
3728:
3724:
3713:
3710:
3702:
3692:
3688:
3682:
3681:
3676:This section
3674:
3670:
3665:
3664:
3658:
3656:
3642:
3637:
3625:
3621:
3616:
3612:
3605:
3601:
3596:
3592:
3585:
3581:
3576:
3569:
3560:
3550:
3546:
3541:
3537:
3534:
3531:
3524:
3520:
3515:
3511:
3504:
3500:
3495:
3486:
3482:
3473:
3469:
3465:
3457:
3453:
3448:
3435:
3432:
3427:
3417:
3413:
3404:
3401:
3398:
3394:
3385:
3378:
3374:
3369:
3356:
3353:
3350:
3344:
3336:
3332:
3326:
3323:
3320:
3316:
3284:
3281:
3278:
3275:
3272:
3267:
3263:
3256:
3253:
3250:
3247:
3238:
3235:
3233:
3227:
3224:
3213:
3210:
3207:
3201:
3198:
3195:
3186:
3183:
3181:
3175:
3172:
3158:
3154:
3153:
3152:
3140:
3129:
3125:
3120:
3105:
3101:
3094:
3091:
3078:
3075:
3072:
3063:
3060:
3057:
3051:
3040:
3034:
3031:
3025:
3019:
3013:
2999:
2993:
2984:
2971:
2965:
2962:
2959:
2953:
2950:
2947:
2938:
2932:
2929:
2923:
2917:
2911:
2903:
2886:
2875:
2871:
2865:
2861:
2856:
2853:In a general
2848:
2832:
2829:
2826:
2821:
2817:
2813:
2806:
2803:
2800:
2795:
2792:
2789:
2777:
2773:
2766:
2764:
2762:
2746:
2727:
2726:central angle
2683:
2679:
2675:
2672:
2667:
2663:
2660:
2654:
2649:
2645:
2642:
2624:
2620:
2617:? This is an
2598:
2594:
2591:
2559:
2555:
2552:
2526:
2518:
2515:
2511:
2504:
2487:
2480:
2475:
2472:
2465:
2462:
2459:
2429:
2422:
2418:
2372:into segment
2361:
2356:
2352:
2348:
2339:
2322:
2307:
2301:
2295:
2290:
2271:
2267:
2263:
2259:
2255:
2245:
2238:
2231:
2201:
2195:
2189:
2186:
2180:
2171:
2168:
2165:
2159:
2150:
2144:
2138:
2132:
2126:
2123:
2120:
2096:
2088:
2084:
2064:
2053:of the ratio
2052:
2035:
2029:
2025:
2011:
2007:
2003:
1998:
1988:
1984:
1980:
1976:
1972:
1968:
1964:
1960:
1956:
1952:
1947:
1945:
1941:
1930:
1919:
1913:
1902:
1893:
1889:
1883:
1875:
1863:
1861:
1858:
1852:
1842:
1839:
1834:
1820:
1817:
1812:
1808:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1781:
1778:
1773:
1769:
1766:
1745:
1742:
1739:
1733:
1730:
1724:
1721:
1696:
1690:
1682:
1666:
1661:
1656:
1650:
1647:
1641:
1635:
1632:
1621:
1616:
1609:
1607:
1602:
1593:
1564:
1561:
1553:
1548:
1545:
1539:
1533:
1526:
1511:
1508:
1505:
1502:
1499:
1493:
1487:
1480:
1479:
1478:
1476:
1472:
1471:complex plane
1467:
1465:
1461:
1451:
1447:
1443:
1435:
1428:
1424:
1419:
1417:
1413:
1407:
1389:
1379:
1376:
1366:
1362:
1358:
1326:
1322:
1318:
1315:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1266:
1260:
1253:
1252:
1251:
1231:
1226:
1211:
1201:
1198:
1187:
1185:
1181:
1177:
1171:
1166:
1162:
1158:
1151:
1139:
1133:
1129:
1125:
1105:
1099:
1096:
1093:
1087:
1083:
1080:
1071:
1065:
1062:
1056:
1050:
1044:
1037:
1036:
1035:
1022:
1015:
1011:
1007:
1003:
999:
991:
986:
983:
980:
976:
973:
969:
967:
964:
961:
958:
956:
952:
949:
947:
944:
942:
939:
937:
936:Line segments
934:
931:
927:
924:
923:
922:
916:
914:
913:are similar.
912:
899:
897:
893:
888:
884:
875:
868:
866:
864:
860:
856:
852:
848:
844:
840:
835:
833:
829:
825:
820:
818:
814:
810:
801:
797:
787:
781:
774:
763:
759:
755:
752:
748:
745:
741:
738:
734:
733:
732:
729:
711:
708:
703:
700:
695:
692:
685:
683:
678:
675:
672:
661:
658:
653:
650:
645:
642:
635:
633:
628:
625:
622:
607:
604:
597:
587:
586:
585:
572:
568:
565:
560:
557:
552:
549:
542:
539:
536:
533:
526:
515:
512:
507:
504:
493:
490:
483:
472:
469:
464:
461:
450:
447:
432:
431:
428:of the other.
427:
423:
422:
421:
408:
397:
394:
389:
386:
375:
372:
365:
354:
351:
346:
343:
332:
329:
322:
311:
308:
303:
300:
289:
286:
271:
270:
263:
256:
249:
242:
235:
228:
222:
221:
220:
219:
215:
214:
213:
210:
208:
204:
200:
195:
188:
178:
176:
173:
168:
160:
156:
153:
149:
145:
141:
137:
133:
120:
116:
110:
106:
100:
96:
90:
86:
82:
80:
76:
72:
68:
64:
60:
56:
52:
43:
37:
33:
19:
4894:
4881:
4869:
4848:
4832:
4813:
4794:
4775:
4754:
4733:
4711:
4689:
4663:
4650:
4616:
4609:
4597:
4585:
4573:
4561:
4549:
4524:
4512:
4500:
4487:
4467:
4457:
4445:
4434:
4425:
4401:
4395:
4386:
4380:
4371:
4365:
4348:
4339:
4329:
4317:
4221:
4208:
4204:adding to it
4199:
4156:
4151:
4133:
4129:
4123:
4120:
4050:
3965:Reflectivity
3964:
3959:
3823:
3817:
3751:
3743:
3738:
3730:
3727:metric space
3720:
3705:
3696:
3685:Please help
3680:verification
3677:
3474:
3467:
3460:
3449:
3386:
3376:
3370:
3308:
3135:
3127:
3123:
3115:
3103:
3099:
3095:
3092:
3000:
2985:
2904:
2869:
2863:
2859:
2855:metric space
2852:
2761:intersection
2708:that joins
2531:from square
2519:
2513:
2509:
2502:
2359:
2354:
2350:
2346:
2323:
2269:
2265:
2261:
2257:
2253:
2243:
2236:
2232:
2094:
1999:
1966:
1948:
1943:
1937:
1928:
1887:
1856:
1850:
1843:
1835:
1692:
1620:tessellation
1600:
1591:
1580:
1468:
1460:affine group
1449:
1445:
1426:
1420:
1415:
1411:
1408:
1360:
1356:
1305:
1188:
1183:
1169:
1164:
1160:
1156:
1131:
1127:
1123:
1120:
1005:
1001:
997:
995:
955:eccentricity
920:
900:
887:proportional
880:
836:
821:
799:
795:
779:
772:
768:
744:transitivity
739:are similar.
730:
608:
606:as follows:
602:
595:
591:
436:
275:
261:
254:
247:
240:
233:
226:
211:
206:
203:proportional
193:
186:
182:
169:
165:
129:
54:
48:
4517:Venema 2006
4505:Venema 2006
4495:(1616–1703)
4493:John Wallis
4450:Jacobs 1974
4418:Venema 2006
4354:Venema 2006
4322:Sibley 1998
3699:August 2018
2868:, an exact
2755:. So point
2745:is a member
2623:orientation
2386:and point
1959:is repeated
1418:change it.
1021:real number
765:proportion.
426:enlargement
92:Translation
67:translation
4934:Categories
4730:Pedoe, Dan
4675:References
4602:Pedoe 1988
4578:Pedoe 1988
4554:Smart 1998
4491:Named for
4334:Stahl 2003
4183:Psychology
4051:Finiteness
3731:similarity
3151:for which
3143:such that
2870:similitude
2629:such that
2338:reflection
2214:by naming
2002:decomposed
1912:decomposed
1006:similitude
998:similarity
966:Catenaries
951:Hyperbolas
901:For given
896:rectangles
792:such that
762:hypotenuse
148:rectangles
112:Reflection
75:reflection
4736:. Dover.
4732:(1988) .
4590:Yale 1968
4566:Yale 1968
4400:Euclid's
4385:Euclid's
4370:Euclid's
4338:Euclid's
4211:July 2021
4165:{2, 3·2}
4103:∞
4061:∀
3975:∀
3937:⇔
3877:∀
3849:≤
3798:≥
3759:∀
3613:⋯
3593:⋅
3538:∘
3535:⋯
3532:∘
3512:∘
3483:μ
3402:∈
3395:∑
3324:∈
3317:⋃
3279:−
3268:∗
3108:is a set
2992:Lipschitz
2890:, called
2827:
2804:
2793:
2684:∘
2668:→
2650:→
2599:→
2578:into ray
2560:→
2406:of ratio
2291:−
2169:∘
2085:) of the
1987:homothety
1796:⋅
1787:⋅
1604:|= 1
1557:¯
1380:∈
1319:∈
1217:→
1189:As a map
1014:bijection
972:logarithm
946:Parabolas
930:congruent
839:synthetic
751:altitudes
689:△
686:≁
670:△
639:△
636:∼
620:△
546:∠
543:≅
531:∠
521:¯
498:¯
478:¯
455:¯
403:¯
380:¯
360:¯
337:¯
317:¯
294:¯
172:congruent
79:congruent
4940:Geometry
4712:Geometry
4710:(1974).
4688:(2005).
4636:61331653
4465:(2008).
4402:Elements
4387:Elements
4372:Elements
4340:Elements
4297:Homoeoid
4226:See also
3735:distance
3723:topology
3659:Topology
3228:′
3176:′
2998:a limit
2990:be a bi-
2902:we have
2874:function
2360:indirect
2006:rotation
1985:under a
1983:pentagon
1882:composed
1770:′
1440:and the
1176:isometry
1163:and the
1034:we have
960:Ellipses
907:regular
883:polygons
857:and the
735:Any two
712:′
704:′
696:′
662:′
654:′
646:′
569:′
561:′
553:′
516:′
508:′
473:′
465:′
398:′
390:′
355:′
347:′
312:′
304:′
144:ellipses
102:Rotation
71:rotation
4136:of the
4126:measure
3373:measure
2759:is the
2724:form a
2615:
2580:
2576:
2541:
2520:How to
2444:Point
2442:
2408:
2358:is the
2344:, then
2336:is the
2320:
2279:
2087:inverse
2079:
2055:
2048:
2014:
2004:into a
1971:segment
1953:into a
1838:volumes
1596:. When
1404:
1369:
1352:
1308:
1244:
1191:
1184:similar
1167:. When
1136:is the
1008:) of a
941:Circles
248:A'B'C',
234:B'A'C',
136:squares
132:circles
122:Scaling
63:scaling
55:similar
4901:
4855:
4820:
4801:
4782:
4763:
4740:
4718:
4696:
4634:
4624:
4475:
4167:where
4079:
3993:
3141:< 1
2885:scalar
2390:into
2324:With "
2095:center
1979:spiral
1967:center
1944:center
1929:center
1888:center
1886:has a
1598:|
1581:where
1354:is an
1306:where
1159:, the
1150:scalar
1148:. The
1121:where
905:, all
892:rhombi
845:, the
603:A'B'C'
262:A'C'B'
194:A'B'C'
4632:S2CID
4309:Notes
2872:is a
2720:and
2712:and
2382:into
2368:like
1975:image
1963:abyss
1918:areas
1695:areas
1475:reals
1423:group
1140:from
1012:is a
926:Lines
911:-gons
826:(see
815:. In
59:shape
4899:ISBN
4853:ISBN
4818:ISBN
4799:ISBN
4780:ISBN
4761:ISBN
4738:ISBN
4716:ISBN
4694:ISBN
4622:ISBN
4473:ISBN
4140:and
4100:<
3725:, a
3454:and
3134:0 ≤
2898:and
2753:BCAT
2738:ABEF
2533:ABEF
2400:ABEF
2396:ACBT
2222:and
1618:The
1585:and
1367:and
1030:and
756:Two
599:and
237:and
201:are
190:and
170:Two
73:and
34:and
4290:on
4206:.
4130:not
3872:and
3721:In
3689:by
3239:0.1
3187:0.1
3008:lim
2818:log
2801:log
2790:log
2728:of
2680:135
2505:= 0
2244:ATB
2237:EFA
1594:≠ 0
1436:of
1172:= 1
1144:to
1004:or
800:DEF
798:~ △
796:ABC
773:ABC
596:ABC
255:ACB
241:ABC
227:BAC
223:If
187:ABC
49:In
4936::
4884:.
4872:.
4684:;
4630:.
4535:^
4410:^
4053::
4047:or
3967::
3820:):
3114:{
3102:,
2862:,
2849:.)
2705:EA
2517:.
2512:=
2503:AK
2375:CT
2365:BF
2353:=
2349:○
2342:CW
2268:○
2264:=
2260:○
2256:=
2218:,
1997:.
1991:–k
1860:.
1712:kh
1708:kb
1466:.
1359:×
996:A
865:.
853:,
849:,
834:.
780:DE
69:,
4907:.
4876:.
4861:.
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4807:.
4788:.
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3925:a
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3919:S
3916:=
3913:)
3910:b
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3904:a
3901:(
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3892:)
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3880:(
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3855:(
3852:S
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3786:a
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3777:,
3774:)
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3768:,
3765:a
3762:(
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3706:(
3701:)
3697:(
3683:.
3643:.
3638:D
3634:)
3626:n
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3617:r
3606:2
3602:s
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3577:r
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3567:)
3564:)
3561:K
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3410:(
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3337:s
3333:f
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3193:(
3190:[
3184:=
3173:z
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3128:S
3126:∈
3124:s
3121:}
3118:s
3116:f
3110:K
3106:)
3104:d
3100:X
3098:(
3079:.
3076:r
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3067:)
3064:y
3061:,
3058:x
3055:(
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3047:)
3044:)
3041:y
3038:(
3035:f
3032:,
3029:)
3026:x
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2996:r
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2972:.
2969:)
2966:y
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2960:x
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2939:y
2936:(
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2927:)
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2915:(
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2864:d
2860:X
2858:(
2833:,
2830:3
2822:2
2814:=
2807:2
2796:3
2757:S
2749:T
2742:S
2734:F
2722:A
2718:E
2714:A
2710:E
2676:+
2673:=
2664:A
2661:P
2655:,
2646:E
2643:P
2627:P
2595:A
2592:S
2556:E
2553:S
2537:S
2529:D
2524:S
2514:K
2510:A
2488:,
2481:2
2476:K
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2466:=
2463:K
2460:A
2450:K
2446:A
2430:.
2423:2
2419:1
2404:I
2392:A
2388:A
2384:B
2380:E
2370:D
2355:I
2351:D
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2326:M
2308:.
2302:2
2296:2
2275:D
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2266:H
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2254:D
2249:S
2242:△
2235:△
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2220:H
2216:R
2202:,
2199:)
2196:F
2193:(
2190:D
2187:=
2184:)
2181:F
2178:(
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2166:H
2163:(
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2157:)
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2151:F
2148:(
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2142:(
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2133:W
2130:(
2127:H
2124:=
2121:T
2111:W
2107:T
2103:F
2099:W
2091:S
2081:(
2065:2
2036:,
2030:2
2026:2
1995:k
1931:.
1920:.
1908:S
1857:k
1851:k
1846:k
1821:.
1818:A
1813:2
1809:k
1805:=
1802:h
1799:k
1793:b
1790:k
1782:2
1779:1
1774:=
1767:A
1746:,
1743:h
1740:b
1734:2
1731:1
1725:=
1722:A
1704:h
1700:b
1683:.
1667:.
1662:5
1657:=
1651:1
1648:h
1642:=
1636:h
1633:5
1601:a
1592:a
1587:b
1583:a
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1562:+
1554:z
1549:a
1546:=
1543:)
1540:z
1537:(
1534:f
1512:b
1509:+
1506:z
1503:a
1500:=
1497:)
1494:z
1491:(
1488:f
1456:S
1452:)
1450:n
1448:(
1446:E
1438:S
1430:S
1390:n
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1377:t
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1336:R
1332:(
1327:n
1323:O
1316:A
1291:,
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1282:x
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1270:)
1267:x
1264:(
1261:f
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1232:,
1227:n
1222:R
1212:n
1207:R
1202::
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1178:(
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1128:x
1126:(
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1103:)
1100:y
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1091:(
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1075:)
1072:y
1069:(
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1060:)
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1054:(
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1048:(
1045:d
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1028:x
1024:r
1017:f
932:)
909:n
903:n
805:F
794:△
790:F
771:△
709:C
701:B
693:A
679:C
676:B
673:A
659:C
651:B
643:A
629:C
626:B
623:A
601:△
594:△
573:.
566:C
558:B
550:A
540:C
537:B
534:A
527:,
513:C
505:B
494:C
491:B
484:=
470:B
462:A
451:B
448:A
409:.
395:C
387:A
376:C
373:A
366:=
352:C
344:B
333:C
330:B
323:=
309:B
301:A
290:B
287:A
260:∠
253:∠
246:∠
239:∠
232:∠
225:∠
192:△
185:△
38:.
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