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2012:
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2446:" is a geometric shape consisting of a semicircle and two orthogonal line segments, such that the length of the shorter segment is equal to the circle radius. Trisection is executed by leaning the end of the tomahawk's shorter segment on one ray, the circle's edge on the other, so that the "handle" (longer segment) crosses the angle's vertex; the trisection line runs between the vertex and the center of the semicircle.
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Hutcheson constructed a cylinder from the angle to be trisected by drawing an arc across the angle, completing it as a circle, and constructing from that circle a cylinder on which a, say, equilateral triangle was inscribed (a 360-degree angle divided in three). This was then "mapped" onto the angle
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The tomahawk produces the same geometric effect as the paper-folding method: the distance between circle center and the tip of the shorter segment is twice the distance of the radius, which is guaranteed to contact the angle. It is also equivalent to the use of an architects L-Ruler
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As is known ... every cubic construction can be traced back to the trisection of the angle and to the multiplication of the cube, that is, the extraction of the third root. I need only to show how these two classical tasks can be solved by means of the right angle
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While a tomahawk is constructible with compass and straightedge, it is not generally possible to construct a tomahawk in any desired position. Thus, the above construction does not contradict the nontrisectibility of angles with ruler and compass alone.
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that used a string instead of a compass and straight edge. A string can be used as either a straight edge (by stretching it) or a compass (by fixing one point and identifying another), but can also wrap around a cylinder, the key to
Hutcheson's solution.
1071:
Many incorrect methods of trisecting the general angle have been proposed. Some of these methods provide reasonable approximations; others (some of which are mentioned below) involve tools not permitted in the classical problem. The mathematician
133:, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. Other techniques were developed by mathematicians over the centuries.
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An angle can be trisected with a device that is essentially a four-pronged version of a compass, with linkages between the prongs designed to keep the three angles between adjacent prongs equal.
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1986:
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Another means to trisect an arbitrary angle by a "small" step outside the Greek framework is via a ruler with two marks a set distance apart. The next construction is originally due to
2871:
395:. The argument below shows that it is impossible to construct a 20° angle. This implies that a 60° angle cannot be trisected, and thus that an arbitrary angle cannot be trisected.
2532:
3302:
1278:(types of folding operations) can construct cubic extensions (cube roots) of given lengths, whereas ruler-and-compass can construct only quadratic extensions (square roots).
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It follows that, given a segment that is defined to have unit length, the problem of angle trisection is equivalent to constructing a segment whose length is the root of a
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The general problem of angle trisection is solvable by using additional tools, and thus going outside of the original Greek framework of compass and straightedge.
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254:
published a proof of the impossibility of classically trisecting an arbitrary angle in 1837. Wantzel's proof, restated in modern terminology, uses the concept of
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744:. There are angles that are not constructible but are trisectible (despite the one-third angle itself being non-constructible). For example,
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are equal, these three parallel lines delimit two equal segments on every other secant line, and in particular on their common perpendicular
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attempts at solution by naive enthusiasts. These "solutions" often involve mistaken interpretations of the rules, or are simply incorrect.
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282:. From a solution to one of these two problems, one may pass to a solution of the other by a compass and straightedge construction. The
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Trisection, like many constructions impossible by ruler and compass, can easily be accomplished by the operations of paper folding, or
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3288:
1866:
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2838:
359:
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straightedge. The diagrams we use show this construction for an acute angle, but it indeed works for any angle up to 180 degrees.
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can be used as a basis for the bisections. An approximation to any degree of accuracy can be obtained in a finite number of steps.
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with real coefficients can be solved geometrically with compass, straightedge, and an angle trisector if and only if it has three
1088:
Trisection can be approximated by repetition of the compass and straightedge method for bisecting an angle. The geometric series
136:
Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of
1298:
which can be used to make an instrument to trisect angles including Kempe's
Trisector and Sylvester's Link Fan or Isoklinostat.
1843:
of a transversal to two parallel lines. This proves the second desired equality, and thus the correctness of the construction.
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4363:
3996:
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For the historical basis of
Wantzel's proof in the earlier work of Ruffini and Abel, and its timing vis-a-vis Galois, see
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to solve for arbitrary angles. However, some special angles can be trisected: for example, it is trivial to trisect a
4208:
3835:
3651:
3626:
2787:"Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas"
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3701:
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which, if drawn on the plane using other methods, can be used to trisect arbitrary angles. Examples include the
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3781:
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3178:
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generated by these numbers. Therefore, any number that is constructible by a sequence of steps is a root of a
129:
It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. For example,
3215:
266:(whose work, written in 1830, was published only in 1846) and did not use the concepts introduced by Galois.
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equal to one-third of a given arbitrary angle (or divide it into three equal angles), using only two tools:
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1418:. This line can be drawn either by using again the right triangular ruler, or by using a traditional
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can be trivially trisected by ignoring the given angle and directly constructing an angle of measure
722:
392:
347:
283:
38:
using tools beyond an unmarked straightedge and a compass. The example shows trisection of any angle
2438:
A tomahawk trisecting an angle. The tomahawk is formed by the thick lines and the shaded semicircle.
1999:
number of equal parts. Archimedes described how to trisect an angle using the
Archimedean spiral in
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is drawn. Then, the markedness of the ruler comes into play: one mark of the ruler is placed at
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Bieberbach's trisection of an angle (in blue) by means of a right triangular ruler (in red)
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339:. This equivalence reduces the original geometric problem to a purely algebraic problem.
2011:
1052:, which is a Fermat prime), this condition becomes the above-mentioned requirement that
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multiplied by the product of one or more distinct Fermat primes, none of which divides
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251:
148:
115:
3041:
Hutcheson, Thomas W. (May 2001). "Dividing Any Angle into Any Number of Equal Parts".
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A long article with many approximations & means going outside the Greek framework
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gives an expression relating the cosines of the original angle and its trisection:
236:
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104:
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is equivalent to constructing two segments such that the ratio of their length is
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sides can be constructed with ruler, compass, and angle trisector if and only if
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equal to one third of a given arbitrary angle, using only two tools: an unmarked
60:
by a ruler with length equal to the radius of the circle, giving trisected angle
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3730:
3436:
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123:
30:
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is a relatively straightforward generalization of the proof given above that a
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Approximate angle trisection as an animation, max. error of the angle ≈ ±4E-8°
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Ludwig
Bieberbach (1932) "Zur Lehre von den kubischen Konstruktionen",
2457:, it can be also used for trisection angles by the method described in
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2980:. The National Council of Teachers of Mathematics. pp. 39–42.
408:. If 60° could be trisected, the degree of a minimal polynomial of
228:
197:
Three problems proved elusive, specifically, trisecting the angle,
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and of the same radius intersects the line supporting the edge in
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243:
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232:
227:
209:
159:
147:
100:
29:
3148:
What is mathematics?: an elementary approach to ideas and methods
2900:
See also
Feedback on this article in vol. 93, March 2009, p. 156.
3284:
1422:. With a similar construction, one can improve the location of
1357:
as its second intersection with the edge. A circle centered at
1402:. It follows that the original angle is trisected by the line
1394:
in such a way that the second leg of the ruler is tangent at
1995:. The spiral can, in fact, be used to divide an angle into
1701:
the first desired equality. On the other hand, the triangle
721:
However, some angles can be trisected. For example, for any
3174:
Geometric problems of antiquity, including angle trisection
3001:, H. Hasse und L. Schlesinger, Band 167 Berlin, p. 142–146
2420:
to be trisected, with a simple proof of similar triangles.
3276:
Hyperbolic trisection and the spectrum of regular polygons
1029:
equal parts with straightedge and compass if and only if
27:
Construction of an angle equal to one third a given angle
3097:"Angle trisection, the heptagon, and the triskaidecagon"
350:
in a single step from some given numbers is a root of a
269:
The problem of constructing an angle of a given measure
262:. However, Wantzel published these results earlier than
3067:
Isaac, Rufus, "Two mathematical papers without words",
2101:. While keeping the ruler (but not the mark) touching
2066:
be the horizontal line in the adjacent diagram. Angle
1076:
has detailed some of these failed attempts in his book
2040:
Any full set of angles on a straight line add to 180°,
1378:
is placed on the drawing in the following manner: one
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2570:
2502:
2332:
2300:
2262:
2223:
1901:
1782:
1719:
1644:
1506:
1446:
1386:; the vertex of its right angle is placed at a point
2155:, thus (by Fact 3 above) each has two equal angles.
2036:
This requires three facts from geometry (at right):
4477:
4355:
4305:
4279:
4201:
4183:
4082:
4075:
3931:
3893:
3710:
3318:
2636:{\displaystyle n=2^{r}3^{s}p_{1}p_{2}\cdots p_{k},}
1426:, by using that it is the intersection of the line
2691:
2635:
2526:
2458:
2374:
2315:
2283:
2241:
1980:
1832:{\displaystyle {\widehat {AEP}}={\widehat {EPD}},}
1831:
1769:{\displaystyle {\widehat {APE}}={\widehat {AEP}}.}
1768:
1694:{\displaystyle {\widehat {EPD}}={\widehat {DPS}},}
1693:
1556:{\displaystyle {\widehat {BPE}}={\widehat {EPD}}.}
1555:
1492:
2853:MacHale, Desmond. "Constructing integer angles",
2538:, using angle trisection by means of the tomahawk
1493:{\displaystyle {\widehat {EPD}}={\widehat {DPS}}}
1266:Mathematics of origami § Trisecting an angle
178:into an arbitrary set of equal segments, to draw
3146:Courant, Richard, Herbert Robbins, Ian Stewart,
2923:. Chapman and Hall Mathematics. pp. g. 58.
1981:{\displaystyle 2x(x^{2}+y^{2})=a(3x^{2}-y^{2}),}
2999:Journal für die reine und angewandte Mathematik
2656:are distinct primes greater than 3 of the form
2131:is drawn to make it obvious that line segments
2074:) is the subject of trisection. First, a point
1321:Journal für die reine und angewandte Mathematik
194:of equal or twice the area of a given polygon.
342:Every rational number is constructible. Every
3296:
2409:Thomas Hutcheson published an article in the
2393:Again, this construction stepped outside the
8:
2872:"Trisecting angles with ruler and compasses"
2794:Journal de Mathématiques Pures et Appliquées
2749:
2747:
2489:An animation of a neusis construction of a
2015:Trisection of the angle using a marked ruler
2043:The sum of angles of any triangle is 180°,
677:, but none of these is a root. Therefore,
235:. The displayed ones are marked — an ideal
4309:
4079:
3303:
3289:
3281:
3087:
3085:
3083:
1349:of the angle to be trisected, centered at
1326:Zur Lehre von den kubischen Konstruktionen
807:, which is a full circle plus the desired
2677:
2667:
2661:
2624:
2611:
2601:
2591:
2581:
2569:
2503:
2501:
2331:
2299:
2261:
2222:
1966:
1953:
1928:
1915:
1900:
1872:Trisection using the Maclaurin trisectrix
1807:
1806:
1784:
1783:
1781:
1744:
1743:
1721:
1720:
1718:
1713:of a circle are equal; this implies that
1669:
1668:
1646:
1645:
1643:
1531:
1530:
1508:
1507:
1505:
1471:
1470:
1448:
1447:
1445:
911:with the product of one or more distinct
765:is such an angle: five angles of measure
609:has degree 3, if it is reducible over by
205:. The problem of angle trisection reads:
2865:
2863:
2484:
2010:
1328:. He states therein (free translation):
1285:
2743:
2143:all have equal length. Now, triangles
1860:Trisection using the Archimedean spiral
1850:
1341:The construction begins with drawing a
118:proved that the problem, as stated, is
4586:Compass and straightedge constructions
4332:Latin translations of the 12th century
2109:. The mark on the circle is labeled
2029:, i.e., that uses tools other than an
1440:One has to prove the angle equalities
1430:and its perpendicular passing through
1084:Approximation by successive bisections
258:, a topic now typically combined with
4062:Straightedge and compass construction
3184:One link of marked ruler construction
2056:meet the third side at the same angle
1420:straightedge and compass construction
1353:on an edge of this angle, and having
354:of degree 2 with coefficients in the
93:straightedge and compass construction
7:
4027:Incircle and excircles of a triangle
3150:, Oxford University Press US, 1996.
2830:History of Mathematics: A Supplement
2459:§ With a right triangular ruler
2113:and the mark on the line is labeled
786:combine to make an angle of measure
466:. Then by the triple-angle formula,
2763:Mathematical Association of America
2527:{\displaystyle {\overline {OA}}=6}
1382:of its right angle passes through
25:
3104:The American Mathematical Monthly
1611:is the intersection of the lines
698:, and the minimal polynomial for
418:would be a power of two. Now let
99:. It concerns construction of an
4544:
4531:
2987:from the original on 2022-10-09.
2401:by using a marked straightedge.
1877:There are certain curves called
1865:
1853:
144:Background and problem statement
3073:48, 1975, p. 198. Reprinted in
2807:from the original on 2022-10-09
2453:As a tomahawk can be used as a
4364:A History of Greek Mathematics
3877:The Quadrature of the Parabola
3189:Another, mentioning Archimedes
2870:McLean, K. Robin (July 2008).
1972:
1943:
1934:
1908:
34:Angles may be trisected via a
1:
2473:With interconnected compasses
2375:{\displaystyle a=c+b=2b+b=3b}
2294:From the last two equations,
1883:trisectrix of Colin Maclaurin
1638:are congruent, and thus that
1302:With a right triangular ruler
1294:There are a number of simple
1045:. In the case of trisection (
907:or the product of a power of
717:Angles which can be trisected
583:, the minimal polynomial for
216:an unmarked straightedge, and
4145:Intersecting secants theorem
3005:. Retrieved on June 2, 2017.
2692:{\displaystyle 2^{t}3^{u}+1}
2513:
1025:radians can be divided into
4140:Intersecting chords theorem
4007:Doctrine of proportionality
1839:since these two angles are
1345:passing through the vertex
4602:
3836:On the Sphere and Cylinder
3789:On the Sizes and Distances
2728:Morley's trisector theorem
2427:
2050:Any two equal sides of an
1398:to the circle centered at
1263:
996:angle is not trisectible.
919:Algebraic characterization
91:is a classical problem of
4538:Ancient Greece portal
4527:
4342:Philosophy of mathematics
4312:
4257:Ptolemy's table of chords
3312:Ancient Greek mathematics
3205:Other means of trisection
2888:10.1017/S0025557200183317
2833:, Springer, p. 130,
2827:Smorynski, Craig (2007),
923:Again, denote the set of
4571:Euclidean plane geometry
4209:Aristarchus's inequality
3782:On Conoids and Spheroids
3017:"Trisection of an Angle"
2971:Yates, Robert C (1942).
2946:"Trisection of an Angle"
2481:Uses of angle trisection
2284:{\displaystyle e+2b=180}
2165:is a straight line, and
1004:For any nonzero integer
174:found means to divide a
4317:Ancient Greek astronomy
4130:Inscribed angle theorem
4120:Greek geometric algebra
3775:Measurement of a Circle
3077:78, April 2005, p. 111.
2857:66, June 1982, 144–145.
2785:Wantzel, P M L (1837).
2242:{\displaystyle e+c=180}
2177:all have equal length,
2078:is drawn at an angle's
1847:With an auxiliary curve
709:So an angle of measure
166:Using only an unmarked
4551:Mathematics portal
4337:Non-Euclidean geometry
4292:Mouseion of Alexandria
4165:Tangent-secant theorem
4115:Geometric mean theorem
4100:Exterior angle theorem
4095:Angle bisector theorem
3799:On Sizes and Distances
3093:Gleason, Andrew Mattei
2974:The Trisection Problem
2693:
2637:
2539:
2528:
2439:
2376:
2317:
2285:
2243:
2082:, one unit apart from
2016:
1982:
1833:
1770:
1695:
1619:. It follows that the
1557:
1494:
1375:right triangular ruler
1311:
1291:
1008:, an angle of measure
1000:Other numbers of parts
971:is reducible over the
939:: An angle of measure
837:, an angle of measure
731:, an angle of measure
248:
240:
224:Proof of impossibility
163:
85:
4239:Pappus's area theorem
4175:Theorem of the gnomon
4052:Quadratrix of Hippias
3975:Circles of Apollonius
3923:Problem of Apollonius
3901:Constructible numbers
3725:Archimedes Palimpsest
2723:Constructible polygon
2694:
2638:
2529:
2488:
2437:
2399:allowed constructions
2377:
2318:
2286:
2244:
2117:. This ensures that
2014:
1983:
1887:Cartesian coordinates
1834:
1771:
1696:
1575:are parallel. As the
1558:
1495:
1309:
1289:
1033:is either a power of
713:cannot be trisected.
621:rational root theorem
543:to be the polynomial
246:
231:
162:has long been solved.
151:
33:
4455:prehistoric counting
4252:Ptolemy's inequality
4193:Apollonius's theorem
4032:Method of exhaustion
4002:Diophantine equation
3992:Circumscribed circle
3809:On the Moving Sphere
3267:sciencenews.org site
3133:on November 5, 2014.
3075:Mathematics Magazine
3070:Mathematics Magazine
3055:10.5951/MT.94.5.0400
2952:on February 25, 2012
2876:Mathematical Gazette
2855:Mathematical Gazette
2718:Constructible number
2660:
2568:
2500:
2330:
2316:{\displaystyle c=2b}
2298:
2260:
2252:Looking at triangle
2221:
2217:From Fact 1) above,
1899:
1780:
1717:
1642:
1504:
1444:
1414:and passing through
1290:Sylvester's Link Fan
1056:not be divisible by
623:, this root must be
284:triple-angle formula
186:, to construct many
172:Greek mathematicians
4581:History of geometry
4541: •
4347:Neusis construction
4267:Spiral of Theodorus
4160:Pythagorean theorem
4105:Euclidean algorithm
4047:Lune of Hippocrates
3916:Squaring the circle
3672:Theon of Alexandria
3347:Aristaeus the Elder
3043:Mathematics Teacher
3023:on November 4, 2013
2430:Tomahawk (geometry)
2412:Mathematics Teacher
2026:Neusis construction
2007:With a marked ruler
203:squaring the circle
190:, and to construct
131:neusis construction
36:neusis construction
4576:Unsolvable puzzles
4234:Menelaus's theorem
4224:Irrational numbers
4037:Parallel postulate
4012:Euclidean geometry
3980:Apollonian circles
3522:Isidore of Miletus
3246:Archimedean Spiral
3003:online-copie (GDZ)
2689:
2647:≥ 0 and where the
2633:
2540:
2524:
2467:Carpenter's Square
2440:
2372:
2313:
2281:
2239:
2052:isosceles triangle
2017:
1993:Archimedean spiral
1978:
1829:
1766:
1691:
1553:
1490:
1312:
1292:
702:is of degree
398:Denote the set of
391:, written 60°) is
362:whose degree is a
360:minimal polynomial
249:
241:
164:
138:pseudomathematical
86:
4558:
4557:
4523:
4522:
4275:
4274:
4262:Ptolemy's theorem
4135:Intercept theorem
3985:Apollonian gasket
3911:Doubling the cube
3884:The Sand Reckoner
3156:978-0-19-510519-3
2944:Jim Loy (2003) .
2930:978-0-412-34550-0
2772:978-0-88385-514-0
2755:Dudley, Underwood
2703:greater than 3).
2536:Andrew M. Gleason
2516:
2424:With a "tomahawk"
2097:and the other at
1891:implicit equation
1823:
1800:
1760:
1737:
1685:
1662:
1547:
1524:
1487:
1464:
1410:perpendicular to
1316:Ludwig Bieberbach
1037:or is a power of
945:may be trisected
344:irrational number
199:doubling the cube
97:Greek mathematics
16:(Redirected from
4593:
4549:
4548:
4536:
4535:
4534:
4310:
4297:Platonic Academy
4244:Problem II.8 of
4214:Crossbar theorem
4170:Thales's theorem
4110:Euclid's theorem
4080:
3997:Commensurability
3958:Axiomatic system
3906:Angle trisection
3871:
3861:
3823:
3813:
3803:
3793:
3769:
3759:
3742:
3305:
3298:
3291:
3282:
3222:2009-10-25) the
3135:
3134:
3132:
3126:. Archived from
3101:
3089:
3078:
3065:
3059:
3058:
3038:
3032:
3031:
3029:
3028:
3019:. Archived from
3012:
3006:
2995:
2989:
2988:
2986:
2979:
2968:
2962:
2961:
2959:
2957:
2948:. Archived from
2941:
2935:
2934:
2909:
2903:
2902:
2867:
2858:
2851:
2845:
2843:
2823:
2817:
2816:
2814:
2812:
2806:
2791:
2782:
2776:
2775:
2751:
2698:
2696:
2695:
2690:
2682:
2681:
2672:
2671:
2642:
2640:
2639:
2634:
2629:
2628:
2616:
2615:
2606:
2605:
2596:
2595:
2586:
2585:
2533:
2531:
2530:
2525:
2517:
2512:
2504:
2381:
2379:
2378:
2373:
2322:
2320:
2319:
2314:
2290:
2288:
2287:
2282:
2248:
2246:
2245:
2240:
2206:
2205:
2203:
2202:
2199:
2196:
2176:
2172:
2168:
2164:
2150:
2146:
2142:
2138:
2134:
2130:
2126:
2116:
2112:
2108:
2104:
2100:
2096:
2092:
2085:
2077:
2073:
2069:
2065:
1987:
1985:
1984:
1979:
1971:
1970:
1958:
1957:
1933:
1932:
1920:
1919:
1869:
1857:
1841:alternate angles
1838:
1836:
1835:
1830:
1825:
1824:
1819:
1808:
1802:
1801:
1796:
1785:
1775:
1773:
1772:
1767:
1762:
1761:
1756:
1745:
1739:
1738:
1733:
1722:
1704:
1700:
1698:
1697:
1692:
1687:
1686:
1681:
1670:
1664:
1663:
1658:
1647:
1637:
1635:
1629:
1627:
1618:
1614:
1610:
1606:
1603:
1596:
1589:
1585:
1581:
1574:
1570:
1566:
1563:The three lines
1562:
1560:
1559:
1554:
1549:
1548:
1543:
1532:
1526:
1525:
1520:
1509:
1499:
1497:
1496:
1491:
1489:
1488:
1483:
1472:
1466:
1465:
1460:
1449:
1433:
1429:
1425:
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1368:
1364:
1360:
1356:
1352:
1348:
1255:
1253:
1251:
1250:
1247:
1244:
1237:
1235:
1234:
1231:
1228:
1221:
1219:
1218:
1215:
1212:
1205:
1203:
1202:
1199:
1196:
1189:
1187:
1186:
1183:
1180:
1172:
1169:
1167:
1166:
1163:
1160:
1153:
1151:
1150:
1147:
1144:
1137:
1135:
1134:
1131:
1128:
1121:
1119:
1118:
1115:
1112:
1104:
1102:
1101:
1098:
1095:
1074:Underwood Dudley
1059:
1055:
1051:
1044:
1040:
1036:
1032:
1028:
1024:
1023:
1022:
1016:
1015:
1007:
995:
984:
970:
944:
932:
925:rational numbers
910:
906:
902:
894:
893:
891:
890:
885:
882:
881:
872:. In contrast,
871:
868:does not divide
867:
859:
858:
856:
855:
850:
847:
846:
836:
832:positive integer
826:
825:
823:
822:
819:
816:
815:
806:
805:
803:
802:
799:
796:
795:
785:
784:
782:
781:
778:
775:
774:
764:
763:
761:
760:
757:
754:
753:
743:
737:
730:
712:
705:
701:
697:
687:
676:
675:
673:
672:
669:
666:
657:
656:
654:
653:
650:
647:
640:
638:
637:
634:
631:
614:
608:
597:
586:
582:
571:
561:
542:
531:
519:
518:
516:
515:
512:
509:
492:
483:
481:
480:
477:
474:
465:
464:
462:
461:
458:
455:
447:
446:
444:
443:
440:
437:
428:
424:
417:
411:
407:
400:rational numbers
383:
382:
380:
379:
376:
373:
337:cubic polynomial
331:
330:
328:
327:
324:
321:
312:
310:
309:
306:
303:
292:
281:
274:
256:field extensions
89:Angle trisection
83:
82:
80:
79:
76:
73:
59:
58:
56:
55:
52:
49:
21:
4601:
4600:
4596:
4595:
4594:
4592:
4591:
4590:
4561:
4560:
4559:
4554:
4543:
4532:
4530:
4519:
4485:Arabian/Islamic
4473:
4462:numeral systems
4351:
4301:
4271:
4219:Heron's formula
4197:
4179:
4071:
4067:Triangle center
4057:Regular polygon
3934:and definitions
3933:
3927:
3889:
3869:
3859:
3821:
3811:
3801:
3791:
3767:
3757:
3740:
3706:
3677:Theon of Smyrna
3322:
3314:
3309:
3207:
3165:
3143:
3141:Further reading
3138:
3130:
3116:10.2307/2323624
3099:
3091:
3090:
3081:
3066:
3062:
3040:
3039:
3035:
3026:
3024:
3015:
3013:
3009:
2996:
2992:
2984:
2977:
2970:
2969:
2965:
2955:
2953:
2943:
2942:
2938:
2931:
2911:
2910:
2906:
2869:
2868:
2861:
2852:
2848:
2841:
2826:
2824:
2820:
2810:
2808:
2804:
2789:
2784:
2783:
2779:
2773:
2753:
2752:
2745:
2741:
2709:
2701:Pierpont primes
2673:
2663:
2658:
2657:
2655:
2620:
2607:
2597:
2587:
2577:
2566:
2565:
2558:regular polygon
2505:
2498:
2497:
2493:with radius of
2483:
2475:
2432:
2426:
2407:
2328:
2327:
2296:
2295:
2258:
2257:
2256:, from Fact 2)
2219:
2218:
2200:
2197:
2192:
2191:
2189:
2184:
2174:
2170:
2166:
2162:
2148:
2144:
2140:
2136:
2132:
2128:
2118:
2114:
2110:
2106:
2102:
2098:
2094:
2090:
2083:
2075:
2071:
2070:(left of point
2067:
2063:
2009:
2003:around 225 BC.
1962:
1949:
1924:
1911:
1897:
1896:
1873:
1870:
1861:
1858:
1849:
1809:
1786:
1778:
1777:
1746:
1723:
1715:
1714:
1702:
1671:
1648:
1640:
1639:
1633:
1631:
1625:
1623:
1621:right triangles
1616:
1612:
1608:
1601:
1594:
1591:
1587:
1583:
1579:
1572:
1568:
1564:
1533:
1510:
1502:
1501:
1473:
1450:
1442:
1441:
1431:
1427:
1423:
1415:
1411:
1407:
1406:, and the line
1403:
1399:
1395:
1391:
1387:
1383:
1366:
1362:
1358:
1354:
1350:
1346:
1304:
1284:
1282:Using a linkage
1276:Huzita's axioms
1268:
1262:
1248:
1245:
1242:
1241:
1239:
1232:
1229:
1226:
1225:
1223:
1216:
1213:
1210:
1209:
1207:
1200:
1197:
1194:
1193:
1191:
1184:
1181:
1178:
1177:
1175:
1174:
1164:
1161:
1158:
1157:
1155:
1148:
1145:
1142:
1141:
1139:
1132:
1129:
1126:
1125:
1123:
1116:
1113:
1110:
1109:
1107:
1099:
1096:
1093:
1092:
1090:
1089:
1086:
1066:
1057:
1053:
1046:
1042:
1038:
1034:
1030:
1026:
1018:
1013:
1011:
1010:
1009:
1005:
1002:
993:
975:
973:field extension
949:
940:
928:
921:
908:
904:
900:
899:if and only if
886:
883:
879:
877:
876:
874:
873:
869:
865:
864:if and only if
851:
848:
844:
842:
841:
839:
838:
834:
820:
817:
813:
812:
811:
809:
808:
800:
797:
793:
791:
790:
788:
787:
779:
776:
772:
770:
769:
767:
766:
758:
755:
751:
749:
748:
746:
745:
739:
732:
726:
719:
710:
703:
699:
693:
678:
670:
667:
664:
663:
661:
659:
651:
648:
645:
644:
642:
635:
632:
629:
628:
626:
624:
610:
599:
588:
587:is a factor of
584:
573:
566:
544:
533:
521:
513:
510:
507:
506:
504:
494:
478:
475:
472:
471:
469:
467:
459:
456:
453:
452:
450:
449:
441:
438:
435:
434:
432:
430:
426:
419:
413:
409:
403:
377:
374:
371:
370:
368:
367:
325:
322:
317:
316:
314:
307:
304:
299:
298:
296:
294:
287:
276:
270:
264:Évariste Galois
226:
170:and a compass,
146:
77:
74:
69:
68:
66:
61:
53:
50:
47:
46:
44:
39:
28:
23:
22:
15:
12:
11:
5:
4599:
4597:
4589:
4588:
4583:
4578:
4573:
4563:
4562:
4556:
4555:
4528:
4525:
4524:
4521:
4520:
4518:
4517:
4512:
4507:
4502:
4497:
4492:
4487:
4481:
4479:
4478:Other cultures
4475:
4474:
4472:
4471:
4470:
4469:
4459:
4458:
4457:
4447:
4446:
4445:
4435:
4434:
4433:
4423:
4422:
4421:
4411:
4410:
4409:
4399:
4398:
4397:
4387:
4386:
4385:
4375:
4374:
4373:
4359:
4357:
4353:
4352:
4350:
4349:
4344:
4339:
4334:
4329:
4327:Greek numerals
4324:
4322:Attic numerals
4319:
4313:
4307:
4303:
4302:
4300:
4299:
4294:
4289:
4283:
4281:
4277:
4276:
4273:
4272:
4270:
4269:
4264:
4259:
4254:
4249:
4241:
4236:
4231:
4226:
4221:
4216:
4211:
4205:
4203:
4199:
4198:
4196:
4195:
4189:
4187:
4181:
4180:
4178:
4177:
4172:
4167:
4162:
4157:
4152:
4150:Law of cosines
4147:
4142:
4137:
4132:
4127:
4122:
4117:
4112:
4107:
4102:
4097:
4091:
4089:
4077:
4073:
4072:
4070:
4069:
4064:
4059:
4054:
4049:
4044:
4042:Platonic solid
4039:
4034:
4029:
4024:
4022:Greek numerals
4019:
4014:
4009:
4004:
3999:
3994:
3989:
3988:
3987:
3982:
3972:
3967:
3966:
3965:
3955:
3954:
3953:
3948:
3937:
3935:
3929:
3928:
3926:
3925:
3920:
3919:
3918:
3913:
3908:
3897:
3895:
3891:
3890:
3888:
3887:
3880:
3873:
3863:
3853:
3850:Planisphaerium
3846:
3839:
3832:
3825:
3815:
3805:
3795:
3785:
3778:
3771:
3761:
3751:
3744:
3734:
3727:
3722:
3714:
3712:
3708:
3707:
3705:
3704:
3699:
3694:
3689:
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3649:
3644:
3639:
3634:
3629:
3624:
3619:
3614:
3609:
3604:
3599:
3594:
3589:
3584:
3579:
3574:
3569:
3564:
3559:
3554:
3549:
3544:
3539:
3534:
3529:
3524:
3519:
3514:
3509:
3504:
3499:
3494:
3489:
3484:
3479:
3474:
3469:
3464:
3459:
3454:
3449:
3444:
3439:
3434:
3429:
3424:
3419:
3414:
3409:
3404:
3399:
3394:
3389:
3384:
3379:
3374:
3369:
3364:
3359:
3354:
3349:
3344:
3339:
3334:
3328:
3326:
3320:Mathematicians
3316:
3315:
3310:
3308:
3307:
3300:
3293:
3285:
3279:
3278:
3273:
3264:
3252:Trisecting via
3249:
3241:Trisecting via
3238:
3216:Trisecting via
3213:
3206:
3203:
3202:
3201:
3196:
3191:
3186:
3181:
3176:
3171:
3169:MathWorld site
3164:
3163:External links
3161:
3160:
3159:
3142:
3139:
3137:
3136:
3110:(3): 185–194.
3095:(March 1988).
3079:
3060:
3049:(5): 400–405.
3033:
3007:
2990:
2963:
2936:
2929:
2904:
2859:
2846:
2839:
2818:
2777:
2771:
2759:The trisectors
2742:
2740:
2737:
2736:
2735:
2730:
2725:
2720:
2715:
2708:
2705:
2688:
2685:
2680:
2676:
2670:
2666:
2651:
2632:
2627:
2623:
2619:
2614:
2610:
2604:
2600:
2594:
2590:
2584:
2580:
2576:
2573:
2544:cubic equation
2523:
2520:
2515:
2511:
2508:
2482:
2479:
2474:
2471:
2428:Main article:
2425:
2422:
2406:
2403:
2384:
2383:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2324:
2312:
2309:
2306:
2303:
2292:
2280:
2277:
2274:
2271:
2268:
2265:
2250:
2238:
2235:
2232:
2229:
2226:
2086:. A circle of
2060:
2059:
2048:
2041:
2008:
2005:
1989:
1988:
1977:
1974:
1969:
1965:
1961:
1956:
1952:
1948:
1945:
1942:
1939:
1936:
1931:
1927:
1923:
1918:
1914:
1910:
1907:
1904:
1875:
1874:
1871:
1864:
1862:
1859:
1852:
1848:
1845:
1828:
1822:
1818:
1815:
1812:
1805:
1799:
1795:
1792:
1789:
1765:
1759:
1755:
1752:
1749:
1742:
1736:
1732:
1729:
1726:
1690:
1684:
1680:
1677:
1674:
1667:
1661:
1657:
1654:
1651:
1552:
1546:
1542:
1539:
1536:
1529:
1523:
1519:
1516:
1513:
1486:
1482:
1479:
1476:
1469:
1463:
1459:
1456:
1453:
1339:
1338:
1303:
1300:
1283:
1280:
1264:Main article:
1261:
1258:
1085:
1082:
1078:The Trisectors
1065:
1062:
1001:
998:
947:if and only if
920:
917:
903:is a power of
718:
715:
615:then it has a
252:Pierre Wantzel
225:
222:
221:
220:
217:
145:
142:
116:Pierre Wantzel
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4598:
4587:
4584:
4582:
4579:
4577:
4574:
4572:
4569:
4568:
4566:
4553:
4552:
4547:
4540:
4539:
4526:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4496:
4493:
4491:
4488:
4486:
4483:
4482:
4480:
4476:
4468:
4465:
4464:
4463:
4460:
4456:
4453:
4452:
4451:
4448:
4444:
4441:
4440:
4439:
4436:
4432:
4429:
4428:
4427:
4424:
4420:
4417:
4416:
4415:
4412:
4408:
4405:
4404:
4403:
4400:
4396:
4393:
4392:
4391:
4388:
4384:
4381:
4380:
4379:
4376:
4372:
4368:
4367:
4366:
4365:
4361:
4360:
4358:
4354:
4348:
4345:
4343:
4340:
4338:
4335:
4333:
4330:
4328:
4325:
4323:
4320:
4318:
4315:
4314:
4311:
4308:
4304:
4298:
4295:
4293:
4290:
4288:
4285:
4284:
4282:
4278:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4247:
4242:
4240:
4237:
4235:
4232:
4230:
4227:
4225:
4222:
4220:
4217:
4215:
4212:
4210:
4207:
4206:
4204:
4200:
4194:
4191:
4190:
4188:
4186:
4182:
4176:
4173:
4171:
4168:
4166:
4163:
4161:
4158:
4156:
4155:Pons asinorum
4153:
4151:
4148:
4146:
4143:
4141:
4138:
4136:
4133:
4131:
4128:
4126:
4125:Hinge theorem
4123:
4121:
4118:
4116:
4113:
4111:
4108:
4106:
4103:
4101:
4098:
4096:
4093:
4092:
4090:
4088:
4087:
4081:
4078:
4074:
4068:
4065:
4063:
4060:
4058:
4055:
4053:
4050:
4048:
4045:
4043:
4040:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3986:
3983:
3981:
3978:
3977:
3976:
3973:
3971:
3968:
3964:
3961:
3960:
3959:
3956:
3952:
3949:
3947:
3944:
3943:
3942:
3939:
3938:
3936:
3930:
3924:
3921:
3917:
3914:
3912:
3909:
3907:
3904:
3903:
3902:
3899:
3898:
3896:
3892:
3886:
3885:
3881:
3879:
3878:
3874:
3872:
3868:
3864:
3862:
3858:
3854:
3852:
3851:
3847:
3845:
3844:
3840:
3838:
3837:
3833:
3831:
3830:
3826:
3824:
3820:
3816:
3814:
3810:
3806:
3804:
3800:
3796:
3794:
3792:(Aristarchus)
3790:
3786:
3784:
3783:
3779:
3777:
3776:
3772:
3770:
3766:
3762:
3760:
3756:
3752:
3750:
3749:
3745:
3743:
3739:
3735:
3733:
3732:
3728:
3726:
3723:
3721:
3720:
3716:
3715:
3713:
3709:
3703:
3700:
3698:
3697:Zeno of Sidon
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3648:
3645:
3643:
3640:
3638:
3635:
3633:
3630:
3628:
3625:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3580:
3578:
3575:
3573:
3570:
3568:
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3545:
3543:
3540:
3538:
3535:
3533:
3530:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3505:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3333:
3330:
3329:
3327:
3325:
3321:
3317:
3313:
3306:
3301:
3299:
3294:
3292:
3287:
3286:
3283:
3277:
3274:
3272:
3268:
3265:
3263:
3262:
3258:
3253:
3250:
3248:
3247:
3242:
3239:
3237:
3236:
3231:
3230:
3226:
3221:
3217:
3214:
3212:
3209:
3208:
3204:
3200:
3199:Geometry site
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3180:
3177:
3175:
3172:
3170:
3167:
3166:
3162:
3157:
3153:
3149:
3145:
3144:
3140:
3129:
3125:
3121:
3117:
3113:
3109:
3105:
3098:
3094:
3088:
3086:
3084:
3080:
3076:
3072:
3071:
3064:
3061:
3056:
3052:
3048:
3044:
3037:
3034:
3022:
3018:
3011:
3008:
3004:
3000:
2994:
2991:
2983:
2976:
2975:
2967:
2964:
2951:
2947:
2940:
2937:
2932:
2926:
2922:
2921:
2919:Galois Theory
2918:
2914:
2908:
2905:
2901:
2897:
2893:
2889:
2885:
2881:
2877:
2873:
2866:
2864:
2860:
2856:
2850:
2847:
2842:
2840:9780387754802
2836:
2832:
2831:
2822:
2819:
2803:
2799:
2795:
2788:
2781:
2778:
2774:
2768:
2764:
2760:
2756:
2750:
2748:
2744:
2738:
2734:
2731:
2729:
2726:
2724:
2721:
2719:
2716:
2714:
2711:
2710:
2706:
2704:
2702:
2686:
2683:
2678:
2674:
2668:
2664:
2654:
2650:
2646:
2630:
2625:
2621:
2617:
2612:
2608:
2602:
2598:
2592:
2588:
2582:
2578:
2574:
2571:
2563:
2559:
2554:
2552:
2549:
2545:
2537:
2521:
2518:
2509:
2506:
2496:
2492:
2487:
2480:
2478:
2472:
2470:
2468:
2462:
2460:
2456:
2451:
2447:
2445:
2436:
2431:
2423:
2421:
2417:
2414:
2413:
2405:With a string
2404:
2402:
2400:
2396:
2391:
2389:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2325:
2310:
2307:
2304:
2301:
2293:
2278:
2275:
2272:
2269:
2266:
2263:
2255:
2251:
2236:
2233:
2230:
2227:
2224:
2216:
2215:
2214:
2212:
2208:
2195:
2187:
2182:
2178:
2160:
2156:
2154:
2125:
2121:
2089:
2081:
2057:
2053:
2049:
2046:
2042:
2039:
2038:
2037:
2034:
2032:
2028:
2027:
2022:
2013:
2006:
2004:
2002:
1998:
1994:
1975:
1967:
1963:
1959:
1954:
1950:
1946:
1940:
1937:
1929:
1925:
1921:
1916:
1912:
1905:
1902:
1895:
1894:
1893:
1892:
1888:
1884:
1880:
1868:
1863:
1856:
1851:
1846:
1844:
1842:
1826:
1820:
1816:
1813:
1810:
1803:
1797:
1793:
1790:
1787:
1776:One has also
1763:
1757:
1753:
1750:
1747:
1740:
1734:
1730:
1727:
1724:
1712:
1708:
1688:
1682:
1678:
1675:
1672:
1665:
1659:
1655:
1652:
1649:
1622:
1605:
1597:
1578:
1577:line segments
1550:
1544:
1540:
1537:
1534:
1527:
1521:
1517:
1514:
1511:
1484:
1480:
1477:
1474:
1467:
1461:
1457:
1454:
1451:
1439:
1435:
1421:
1381:
1377:
1376:
1370:
1344:
1336:
1331:
1330:
1329:
1327:
1323:
1322:
1318:published in
1317:
1308:
1301:
1299:
1297:
1288:
1281:
1279:
1277:
1273:
1267:
1260:Using origami
1259:
1257:
1171:
1083:
1081:
1079:
1075:
1069:
1064:Other methods
1063:
1061:
1049:
1021:
999:
997:
991:
986:
982:
978:
974:
968:
964:
960:
956:
952:
948:
943:
938:
934:
931:
926:
918:
916:
914:
913:Fermat primes
898:
897:constructible
889:
863:
854:
833:
828:
742:
736:
729:
724:
723:constructible
716:
714:
707:
696:
691:
685:
681:
622:
618:
617:rational root
613:
606:
602:
595:
591:
580:
576:
572:is a root of
569:
563:
559:
555:
551:
547:
540:
536:
529:
525:
502:
498:
491:
487:
422:
416:
406:
401:
396:
394:
393:constructible
390:
386:
365:
361:
357:
353:
349:
348:constructible
345:
340:
338:
333:
320:
302:
293: =
291:
285:
280:
273:
267:
265:
261:
260:Galois theory
257:
253:
245:
238:
234:
230:
223:
218:
215:
214:
213:
211:
208:Construct an
206:
204:
200:
195:
193:
189:
185:
184:bisect angles
181:
177:
173:
169:
161:
158:
154:
150:
143:
141:
139:
134:
132:
127:
125:
121:
117:
112:
110:
106:
102:
98:
94:
90:
72:
64:
42:
37:
32:
19:
4542:
4529:
4371:Thomas Heath
4362:
4245:
4229:Law of sines
4085:
4017:Golden ratio
3905:
3882:
3875:
3866:
3860:(Theodosius)
3856:
3848:
3841:
3834:
3827:
3818:
3808:
3802:(Hipparchus)
3798:
3788:
3780:
3773:
3764:
3754:
3746:
3741:(Apollonius)
3737:
3729:
3717:
3692:Zeno of Elea
3452:Eratosthenes
3442:Dionysodorus
3255:
3244:
3233:
3223:
3179:Some history
3147:
3128:the original
3107:
3103:
3074:
3068:
3063:
3046:
3042:
3036:
3025:. Retrieved
3021:the original
3010:
2998:
2993:
2973:
2966:
2954:. Retrieved
2950:the original
2939:
2920:
2917:
2913:Stewart, Ian
2907:
2899:
2879:
2875:
2854:
2849:
2829:
2821:
2809:. Retrieved
2797:
2793:
2780:
2758:
2652:
2648:
2644:
2561:
2555:
2541:
2495:circumcircle
2476:
2463:
2452:
2448:
2441:
2418:
2410:
2408:
2392:
2385:
2253:
2209:
2193:
2185:
2179:
2157:
2127:. A radius
2123:
2119:
2061:
2044:
2035:
2030:
2024:
2018:
1996:
1990:
1879:trisectrices
1876:
1709:, since all
1599:
1592:
1437:
1436:
1390:on the line
1373:
1371:
1340:
1333:
1325:
1319:
1313:
1293:
1269:
1087:
1077:
1070:
1067:
1047:
1019:
1003:
987:
980:
976:
966:
962:
958:
954:
950:
941:
935:
929:
922:
896:
887:
861:
852:
829:
740:
734:
727:
720:
708:
694:
683:
679:
611:
604:
600:
593:
589:
578:
574:
567:
564:
557:
553:
549:
545:
538:
534:
527:
523:
500:
496:
489:
485:
425:. Note that
420:
414:
404:
397:
366:. The angle
364:power of two
341:
334:
318:
300:
289:
278:
271:
268:
250:
239:is un-marked
237:straightedge
207:
196:
168:straightedge
165:
135:
128:
113:
105:straightedge
88:
87:
70:
62:
40:
4438:mathematics
4246:Arithmetica
3843:Ostomachion
3812:(Autolycus)
3731:Arithmetica
3507:Hippocrates
3437:Dinostratus
3422:Dicaearchus
3352:Aristarchus
3232:; see also
2882:: 320–323.
2800:: 366–372.
2534:, based on
2390:is proved.
2326:Therefore,
2023:, called a
1885:, given in
862:trisectible
690:irreducible
598:. Because
124:right angle
95:of ancient
4565:Categories
4490:Babylonian
4390:arithmetic
4356:History of
4185:Apollonius
3870:(Menelaus)
3829:On Spirals
3748:Catoptrics
3687:Xenocrates
3682:Thymaridas
3667:Theodosius
3652:Theaetetus
3632:Simplicius
3622:Pythagoras
3607:Posidonius
3592:Philonides
3552:Nicomachus
3547:Metrodorus
3537:Menaechmus
3492:Hipparchus
3482:Heliodorus
3432:Diophantus
3417:Democritus
3397:Chrysippus
3367:Archimedes
3362:Apollonius
3332:Anaxagoras
3324:(timeline)
3235:Trisectrix
3027:2013-11-04
2739:References
2733:Trisectrix
2455:set square
2181:Conclusion
2159:Hypothesis
2021:Archimedes
2001:On Spirals
619:. By the
352:polynomial
219:a compass.
182:lines, to
120:impossible
18:Trisection
3951:Inscribed
3711:Treatises
3702:Zenodorus
3662:Theodorus
3637:Sosigenes
3582:Philolaus
3567:Oenopides
3562:Nicoteles
3557:Nicomedes
3517:Hypsicles
3412:Ctesibius
3402:Cleomedes
3387:Callippus
3372:Autolycus
3357:Aristotle
3337:Anthemius
3269:on using
3261:Nicomedes
2896:126351853
2713:Bisection
2618:⋯
2514:¯
2395:framework
2153:isosceles
2031:un-marked
1960:−
1821:^
1798:^
1758:^
1735:^
1707:isosceles
1683:^
1660:^
1545:^
1522:^
1485:^
1462:^
1324:his work
1314:In 1932,
570:= cos 20°
532:. Define
423:= cos 20°
346:that is
288:cos
277:cos
247:Compasses
157:arbitrary
153:Bisection
114:In 1837,
4515:Japanese
4500:Egyptian
4443:timeline
4431:timeline
4419:timeline
4414:geometry
4407:timeline
4402:calculus
4395:timeline
4383:timeline
4086:Elements
3932:Concepts
3894:Problems
3867:Spherics
3857:Spherics
3822:(Euclid)
3768:(Euclid)
3765:Elements
3758:(Euclid)
3719:Almagest
3627:Serenus
3602:Porphyry
3542:Menelaus
3497:Hippasus
3472:Eutocius
3447:Domninus
3342:Archytas
3257:Conchoid
3220:Archived
3014:Jim Loy
2982:Archived
2956:30 March
2915:(1989).
2802:Archived
2757:(1994),
2707:See also
2491:heptagon
2444:tomahawk
2386:and the
2183:: angle
2161:: Given
1991:and the
1711:radiuses
1607:, where
1372:Now the
1296:linkages
692:over by
313:− 3 cos
188:polygons
180:parallel
4495:Chinese
4450:numbers
4378:algebra
4306:Related
4280:Centers
4076:Results
3946:Central
3617:Ptolemy
3612:Proclus
3577:Perseus
3532:Marinus
3512:Hypatia
3502:Hippias
3477:Geminus
3467:Eudoxus
3457:Eudemus
3427:Diocles
3271:origami
3225:limacon
3124:2323624
2811:3 March
2645:r, s, k
2388:theorem
2204:
2190:
1889:by the
1590:. Thus
1272:origami
1252:
1240:
1236:
1224:
1220:
1208:
1204:
1192:
1188:
1176:
1168:
1156:
1152:
1140:
1136:
1124:
1120:
1108:
1103:
1091:
1017:⁄
937:Theorem
892:
875:
857:
840:
824:
810:
804:
789:
783:
768:
762:
747:
700:cos 20°
674:
662:
655:
643:
639:
627:
585:cos 20°
530:− 1 = 0
520:. Thus
517:
505:
493:and so
482:
470:
463:
451:
445:
433:
427:cos 60°
410:cos 20°
389:degrees
385:radians
381:
369:
329:
315:
311:
297:
192:squares
109:compass
81:
67:
57:
45:
4510:Indian
4287:Cyrene
3819:Optics
3738:Conics
3657:Theano
3647:Thales
3642:Sporus
3587:Philon
3572:Pappus
3462:Euclid
3392:Carpus
3382:Bryson
3229:Pascal
3154:
3122:
2927:
2894:
2837:
2769:
2699:(i.e.
2643:where
2173:, and
2139:, and
2088:radius
1571:, and
1438:Proof:
1343:circle
965:− cos(
830:For a
725:angle
565:Since
295:4 cos
233:Rulers
201:, and
160:angles
107:and a
4505:Incan
4426:logic
4202:Other
3970:Chord
3963:Axiom
3941:Angle
3597:Plato
3487:Heron
3407:Conon
3131:(PDF)
3120:JSTOR
3100:(PDF)
2985:(PDF)
2978:(PDF)
2892:S2CID
2805:(PDF)
2796:. 1.
2790:(PDF)
2560:with
2551:roots
2211:Proof
2054:will
1634:'
1626:'
1602:'
1595:'
1335:hook.
990:proof
979:(cos(
957:) = 4
625:±1, ±
552:) = 8
412:over
356:field
210:angle
101:angle
43:>
4467:list
3755:Data
3527:Leon
3377:Bion
3254:the
3152:ISBN
2958:2012
2925:ISBN
2835:ISBN
2813:2014
2767:ISBN
2548:real
2151:are
2147:and
2062:Let
1630:and
1615:and
1582:and
1500:and
1365:and
988:The
468:cos
431:cos
387:(60
176:line
4369:by
4083:In
3259:of
3243:an
3227:of
3112:doi
3051:doi
2884:doi
2469:).
2442:A "
2397:of
2279:180
2254:BCD
2237:180
2149:BCD
2145:ABC
2080:ray
2045:and
1997:any
1705:is
1703:PAE
1380:leg
1254:+ ⋯
1173:or
1170:+ ⋯
1165:256
1050:= 3
994:60°
961:− 3
927:by
895:is
860:is
711:60°
688:is
658:or
641:, ±
560:− 1
556:− 6
526:− 6
499:− 3
488:− 3
484:= 4
402:by
155:of
4567::
3118:.
3108:95
3106:.
3102:.
3082:^
3047:94
3045:.
2898:.
2890:.
2880:92
2878:.
2874:.
2862:^
2792:.
2765:,
2761:,
2746:^
2556:A
2553:.
2542:A
2461:.
2291:°.
2249:°.
2213::
2207:.
2188:=
2175:CD
2171:BC
2169:,
2167:AB
2163:AD
2141:CD
2137:BC
2135:,
2133:AB
2129:BC
2124:AB
2122:=
2120:CD
2091:AB
1632:PD
1624:PD
1617:SE
1613:PD
1609:D'
1598:=
1593:SD
1588:SE
1584:PA
1580:OP
1573:AE
1569:PD
1567:,
1565:OS
1434:.
1428:SE
1412:SE
1408:PD
1404:PE
1392:PC
1369:.
1274:.
1249:16
1238:−
1222:+
1206:−
1190:=
1154:+
1149:64
1138:+
1133:16
1122:+
1105:=
1080:.
1060:.
985:.
983:))
933:.
915:.
827:.
792:15
706:.
562:.
503:=
448:=
429:=
332:.
126:.
111:.
65:=
48:3π
3304:e
3297:t
3290:v
3218:(
3158:.
3114::
3057:.
3053::
3030:.
2960:.
2933:.
2886::
2844:.
2815:.
2798:2
2687:1
2684:+
2679:u
2675:3
2669:t
2665:2
2653:i
2649:p
2631:,
2626:k
2622:p
2613:2
2609:p
2603:1
2599:p
2593:s
2589:3
2583:r
2579:2
2575:=
2572:n
2562:n
2522:6
2519:=
2510:A
2507:O
2465:(
2382:.
2370:b
2367:3
2364:=
2361:b
2358:+
2355:b
2352:2
2349:=
2346:b
2343:+
2340:c
2337:=
2334:a
2323:.
2311:b
2308:2
2305:=
2302:c
2276:=
2273:b
2270:2
2267:+
2264:e
2234:=
2231:c
2228:+
2225:e
2201:3
2198:/
2194:a
2186:b
2115:D
2111:C
2107:l
2103:A
2099:B
2095:A
2084:B
2076:A
2072:B
2068:a
2064:l
2058:.
2047:,
1976:,
1973:)
1968:2
1964:y
1955:2
1951:x
1947:3
1944:(
1941:a
1938:=
1935:)
1930:2
1926:y
1922:+
1917:2
1913:x
1909:(
1906:x
1903:2
1827:,
1817:D
1814:P
1811:E
1804:=
1794:P
1791:E
1788:A
1764:.
1754:P
1751:E
1748:A
1741:=
1731:E
1728:P
1725:A
1689:,
1679:S
1676:P
1673:D
1666:=
1656:D
1653:P
1650:E
1636:E
1628:S
1604:E
1600:D
1551:.
1541:D
1538:P
1535:E
1528:=
1518:E
1515:P
1512:B
1481:S
1478:P
1475:D
1468:=
1458:D
1455:P
1452:E
1432:A
1424:E
1416:P
1400:A
1396:E
1388:S
1384:O
1367:O
1363:A
1359:P
1355:B
1351:A
1347:P
1337:"
1332:"
1246:/
1243:1
1233:8
1230:/
1227:1
1217:4
1214:/
1211:1
1201:2
1198:/
1195:1
1185:3
1182:/
1179:1
1162:/
1159:1
1146:/
1143:1
1130:/
1127:1
1117:4
1114:/
1111:1
1100:3
1097:/
1094:1
1058:3
1054:N
1048:n
1043:N
1039:2
1035:2
1031:n
1027:n
1020:N
1014:π
1012:2
1006:N
981:θ
977:Q
969:)
967:θ
963:t
959:t
955:t
953:(
951:q
942:θ
930:Q
909:2
905:2
901:N
888:N
884:/
880:π
878:2
870:N
866:3
853:N
849:/
845:π
843:2
835:N
821:7
818:/
814:π
801:7
798:/
794:π
780:7
777:/
773:π
771:3
759:7
756:/
752:π
750:3
741:θ
735:θ
733:3
728:θ
704:3
695:Q
686:)
684:t
682:(
680:p
671:8
668:/
665:1
660:±
652:4
649:/
646:1
636:2
633:/
630:1
612:Q
607:)
605:t
603:(
601:p
596:)
594:t
592:(
590:p
581:)
579:t
577:(
575:p
568:x
558:t
554:t
550:t
548:(
546:p
541:)
539:t
537:(
535:p
528:x
524:x
522:8
514:2
511:/
508:1
501:x
497:x
495:4
490:x
486:x
479:3
476:/
473:π
460:2
457:/
454:1
442:3
439:/
436:π
421:x
415:Q
405:Q
378:3
375:/
372:π
326:3
323:/
319:θ
308:3
305:/
301:θ
290:θ
279:θ
272:θ
84:.
78:3
75:/
71:θ
63:φ
54:4
51:/
41:θ
20:)
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