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1136:. This is not only simpler but it requires no additional construction at all. The method of proof is to apply side-angle-side to the triangle and its mirror image. More modern authors, in imitation of the method of proof given for the previous proposition have described this as picking up the triangle, turning it over and laying it down upon itself. This method is lampooned by
1955:-style writing. Typical examples are ending a section by telling what the next section is about, without bothering to explain why the topics are related, expanding a casual mention into a detailed treatment, or finding a contrived connection between the topics (e.g. "We bought some red wine; speaking of red liquids, tomorrow is the World Blood Donor Day").
316:
There has been much speculation and debate as to why Euclid added the second conclusion to the theorem, given that it makes the proof more complicated. One plausible explanation, given by
Proclus, is that the second conclusion can be used in possible objections to the proofs of later propositions
1889:
contains the passage "Quot
Euclidis discipulos retrojecit Elefuga quasi scopulos eminens et abruptus, qui nullo scalarum suffragio scandi posset! Durus, inquiunt, est his sermo; quis potest eum audire?", which compares the theorem to a steep cliff that no ladder may help scale and asks how many
1304:. This is simpler than Euclid's proof, but Euclid does not present the construction of an angle bisector until proposition 9. So the order of presentation of Euclid's propositions would have to be changed to avoid the possibility of circular reasoning.
312:
points out, Euclid never uses the second conclusion and his proof can be simplified somewhat by drawing the auxiliary lines to the sides of the triangle instead, the rest of the proof proceeding in more or less the same way.
231:ذُو ٱلْقَرْنَيْن, meaning "the owner of the two horns", because diagrams of the theorem showed two smaller squares like horns at the top of the figure. That term has similarly been used as a metaphor for a dilemma. The name
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wrote a humorous poem called "Pons asinorum" where a geometry class assails the theorem as a company of soldiers might charge a fortress; the battle was not without casualties.
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includes a second conclusion that if the equal sides of the triangle are extended below the base, then the angles between the extensions and the base are also equal. Euclid's
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is the angle between the two vectors, the conclusion of this inner product space form of the theorem is equivalent to the statement about equality of angles.
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The First Six Books of The
Elements of Euclid in which Coloured Diagrams and Symbols are Used Instead of Letters for the Greater Ease of Learners
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169:, referring to the "asses' bridge's" ability to separate capable and incapable reasoners. Its first known usage in this context was in 1645.
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1947:, is used as an awkward transition between them. In serious text, it is considered a stylistic error, since it belongs properly to the
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1943:
is a literary technique where a tenuous, even contrived connection between two arguments or topics, which is almost but not quite a
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325:, which says that given two triangles for which two pairs of corresponding sides and their included angles are respectively
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A. F. West & H. D. Thompson "On
Dulcarnon, Elefuga And Pons Asinorum as Fanciful Names For Geometrical Propositions"
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1479:, Garfield arrived at the proof "in mathematical amusements and discussions with other members of congress."
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1897:, in both its meanings as a bridge and as a test, is used as a metaphor for finding the middle term of a
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The
Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities
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of the intelligence of the reader and functions as a "bridge" to the harder propositions that follow.
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must be used instead of side-angle-side, and side-side-side is not given by Euclid until later in the
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has two meanings – it can describe either a contrived connection between two topics or a mnemonic.
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might play a similar role, as a benchmark indicating whether someone could become a first-class
16:
Statement that the angles opposite the equal sides of an isosceles triangle are themselves equal
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208:"flight", that is "flight of the wretches". Though this etymology is dubious, it is echoed in
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where Euclid does not cover every case. The proof relies heavily on what is today called
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wrote that
Garfield's trapezoid work was "really a very clever proof." According to the
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Jaakko
Hintikka, "On Creativity in Reasoning", in Ake E. Andersson, N.E. Sahlin, eds.,
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2009:(which he was not aware of) by simulating what a mechanical theorem prover might do.
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involves drawing auxiliary lines to these extensions. But, as Euclid's commentator
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Prime
Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
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1971:
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program discovered an original and more elegant proof of this theorem. In fact,
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185:. But the more popular explanation is that it is the first real test in the
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3276:
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2701:
2129:. 500 Fifth Street, NW, Washington D.C. 20001: Joseph Henry Press. p.
1150:" because it apparently requires the triangle to be in two places at once.
221:
was given to the 47th proposition of Book I of Euclid, better known as the
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2204:
Michael A.B. Deakin, "From Pappus to Today: The
History of a Proof",
1898:
217:
2426:(1866: Longmans, Green, Reader, and Dyer) Book 2, Chapter 16, p. 261
2379:"Mathematical Treasure: Garfield's Proof of the Pythagorean Theorem"
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2601:
2469:
Jeremy
Bernstein, "Profiles: A.I." (interview with Marvin Minsky),
1967:
1952:
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developed a proof using the trapezoid, which was published in the
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has itself occasionally been applied to the Pythagorean theorem.
45:
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181:, the simplest being that the diagram used resembles a physical
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1307:
The proof proceeds as follows: As before, let the triangle be
332:
Proclus' variation of Euclid's proof proceeds as follows: Let
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192:
Another medieval term for the isosceles triangle theorem was
49:
103:
1997:
A persistent piece of mathematical folklore claims that an
106:
74:
1796:{\displaystyle \|x-z\|^{2}=\|x\|^{2}-2x\cdot z+\|z\|^{2}}
94:
88:
155:
are equal, then the sides opposite them are also equal.
1878:
as a metaphor for a test of critical thinking include:
2110:. Vol. 2. Ginn & Co. p. 284, footnote 1.
212:
use of the term "flemyng of wreches" for the theorem.
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by a third application of side-angle-side. Therefore
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836:. This sets up another pair of congruent triangles,
137:. The theorem appears as Proposition 5 of Book 1 in
71:
165:for a problem or challenge which acts as a test of
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1132:Proclus gives a much shorter proof attributed to
3822:
1855:{\displaystyle x\cdot z=\|x\|\|z\|\cos \theta ,}
1067:{\displaystyle \triangle BDC\cong \triangle CEB}
875:{\displaystyle \triangle DBE\cong \triangle ECD}
659:{\displaystyle \triangle BAE\cong \triangle CAD}
1296:A standard textbook method is to construct the
1165:being the equal sides. Consider the triangles
177:There are two common explanations for the name
1420:are congruent. It follows that the angles at
1177:is considered a second triangle with vertices
365:be an isosceles triangle with congruent sides
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2556:
1966:('little bridge of asses') is the word for a
241:supposedly once suggested that understanding
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1521:. Unsourced material may be challenged and
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1482:
1115:{\displaystyle \angle CBD\cong \angle BCE}
1019:{\displaystyle \angle BDC\cong \angle CEB}
971:{\displaystyle \angle BED\cong \angle CDE}
923:{\displaystyle \angle BDE\cong \angle CED}
800:. By subtracting congruent line segments,
755:{\displaystyle \angle ADC\cong \angle AEB}
707:{\displaystyle \angle ABE\cong \angle ACD}
2284:(1876 Libr. de Firmin-Didot et Cie) p. 14
2200:
2198:
1890:would-be geometers have been turned away.
1541:Learn how and when to remove this message
1412:, so, applying side-angle-side, triangle
2403:
1553:The isosceles triangle theorem holds in
1287:
281:
265:
20:
2533:D. E. Joyce's presentation of Euclid's
321:(SAS), the previous proposition in the
3823:
3592:Latin translations of the 12th century
2332:
2195:
2005:recounts that he had rediscovered the
884:, again by side-angle-side. Therefore
3322:Straightedge and compass construction
2544:
2407:The Poetical Works of Thomas Campbell
2385:from the original on December 6, 2021
2103:
2086:
2084:
2082:
2080:
2027:
1869:
1405:{\displaystyle \angle BAX=\angle CAX}
257:
56:are themselves equal is known as the
3287:Incircle and excircles of a triangle
2023:
2021:
1519:adding citations to reliable sources
1486:
620:. By side-angle-side, the triangles
329:, then the triangles are congruent.
1240:, so by side-angle-side, triangles
980:. By subtracting congruent angles,
13:
2381:. Mathematical Assoc. of America.
2376:
2293:
2077:
1993:Artificial intelligence proof myth
1390:
1375:
1326:
1319:. Construct the angle bisector of
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2485:Euclid, commentary and trans. by
2092:The Princeton University bulletin
2018:
1368:is equal to itself. Furthermore,
1273:{\displaystyle \angle B=\angle C}
151:is also true: if two angles of a
3851:Theorems about special triangles
3804:
3791:
2300:New England Journal of Education
1702:{\displaystyle \|x-z\|=\|y-z\|.}
1565:. In such spaces, given vectors
1491:
1468:New England Journal of Education
133:", or more descriptively as the
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2463:
2450:
2429:
2424:Principles of Political Economy
2414:
2397:
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2234:
2221:
1457:In 1876, while a member of the
1434:uses a similar construction in
539:. Draw auxiliary line segments
52:opposite the equal sides of an
3624:A History of Greek Mathematics
3137:The Quadrature of the Parabola
2182:
2173:
2164:
2097:
2064:
2058:Merriam-Webster.com Dictionary
2046:
1248:are congruent. In particular,
1189:corresponding respectively to
1157:be an isosceles triangle with
1:
2479:
1153:The proof is as follows: Let
356:{\displaystyle \triangle ABC}
3405:Intersecting secants theorem
2473:December 14, 1981, p. 50-126
2437:The Complexity of Creativity
2344:. Wiley & Sons. p.
2229:Euclid and his Modern Rivals
2104:Smith, David Eugene (1925).
2032:. Taschen. pp. Page 5.
1649:{\displaystyle \|x\|=\|y\|,}
1143:Euclid and his Modern Rivals
172:
7:
3400:Intersecting chords theorem
3267:Doctrine of proportionality
2188:For example F. Cuthbertson
2006:
1970:. The same is true for the
1577:, the theorem says that if
1446:. The proof is similar but
1201:in the original triangle.
503:to make congruent segments
10:
3867:
3096:On the Sphere and Cylinder
3049:On the Sizes and Distances
2227:Charles Lutwidge Dodgson,
2145:first-class mathematician.
1341:{\displaystyle \angle BAC}
1124:, which was to be proved.
827:{\displaystyle BD\cong CE}
791:{\displaystyle BE\cong CD}
530:{\displaystyle AD\cong AE}
401:. Pick an arbitrary point
392:{\displaystyle AB\cong AC}
300:Euclid's statement of the
135:isosceles triangle theorem
3798:Ancient Greece portal
3787:
3737:
3615:
3602:Philosophy of mathematics
3572:
3565:
3539:
3517:Ptolemy's table of chords
3461:
3443:
3342:
3335:
3191:
3153:
2970:
2578:
2572:Ancient Greek mathematics
2458:Mathematics in Management
2404:Campbell, Thomas (1864).
2121:Derbyshire, John (2003).
2094:Vol. 3 No. 4 (1891) p. 84
1283:
1127:
452:and then construct point
252:
3469:Aristarchus's inequality
3042:On Conoids and Spheroids
2492:Vol. 1 (1908 Cambridge)
2460:, 1966, quoted in Deakin
2258:For example J.M. Wilson
2249:Heath p. 254 for section
2206:The Mathematical Gazette
2156:: CS1 maint: location (
2012:
1471:. Mathematics historian
1217:{\displaystyle \angle A}
129:), Latin for "bridge of
3841:Latin words and phrases
3577:Ancient Greek astronomy
3390:Inscribed angle theorem
3380:Greek geometric algebra
3035:Measurement of a Circle
2240:Following Proclus p. 54
2211::467:6-11 (March 1990)
2179:Following Proclus p. 53
1999:artificial intelligence
1949:stream of consciousness
1608:{\displaystyle x+y+z=0}
1483:In inner product spaces
3831:History of mathematics
3811:Mathematics portal
3597:Non-Euclidean geometry
3552:Mouseion of Alexandria
3425:Tangent-secant theorem
3375:Geometric mean theorem
3360:Exterior angle theorem
3355:Angle bisector theorem
3059:On Sizes and Distances
2498:Euclid, commentary by
2107:History Of Mathematics
2072:History of Mathematics
2028:Byrne, Oliver (1847).
1904:The 18th-century poet
1856:
1797:
1703:
1650:
1609:
1459:United States Congress
1442:to be the midpoint of
1406:
1348:and extend it to meet
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3499:Pappus's area theorem
3435:Theorem of the gnomon
3312:Quadratrix of Hippias
3235:Circles of Apollonius
3183:Problem of Apollonius
3161:Constructible numbers
2985:Archimedes Palimpsest
2282:Éléments de géométrie
1857:
1798:
1704:
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1610:
1436:Éléments de géométrie
1407:
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3715:prehistoric counting
3512:Ptolemy's inequality
3453:Apollonius's theorem
3292:Method of exhaustion
3262:Diophantine equation
3252:Circumscribed circle
3069:On the Moving Sphere
2502:, ed. and trans. by
1807:
1715:
1660:
1619:
1581:
1555:inner product spaces
1515:improve this section
1372:
1323:
1252:
1224:is equal to itself,
1205:
1134:Pappus of Alexandria
1082:
1034:
986:
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239:Carl Friedrich Gauss
204:"misery", and Latin
196:which, according to
3836:Elementary geometry
3801: •
3607:Neusis construction
3527:Spiral of Theodorus
3420:Pythagorean theorem
3365:Euclidean algorithm
3307:Lune of Hippocrates
3176:Squaring the circle
2932:Theon of Alexandria
2607:Aristaeus the Elder
2354:1994muaa.book.....D
2262:(1878 Oxford) p. 20
2260:Elementary geometry
2074:(1958 Dover) p. 284
1883:Richard Aungerville
1461:, future President
225:, after the Arabic
223:Pythagorean theorem
200:, comes from Greek
3846:Euclidean geometry
3494:Menelaus's theorem
3484:Irrational numbers
3297:Parallel postulate
3272:Euclidean geometry
3240:Apollonian circles
2782:Isidore of Miletus
2294:G., J. A. (1876).
2192:(1876 Oxford) p. 7
2190:Primer of geometry
1870:Metaphorical usage
1852:
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611:{\displaystyle DE}
608:
584:{\displaystyle DC}
581:
557:{\displaystyle BE}
554:
527:
494:{\displaystyle AC}
491:
464:
443:{\displaystyle AB}
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258:Euclid and Proclus
54:isosceles triangle
38:
33:'s edition of the
3818:
3817:
3783:
3782:
3535:
3534:
3522:Ptolemy's theorem
3395:Intercept theorem
3245:Apollonian gasket
3171:Doubling the cube
3144:The Sand Reckoner
2231:Act I Scene II §6
2170:Heath pp. 251–255
1551:
1550:
1543:
1463:James A. Garfield
1146:, calling it an "
467:{\displaystyle E}
416:{\displaystyle D}
298:
297:
276:
275:
167:critical thinking
3858:
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3808:
3796:
3795:
3794:
3570:
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3557:Platonic Academy
3504:Problem II.8 of
3474:Crossbar theorem
3430:Thales's theorem
3370:Euclid's theorem
3340:
3339:
3257:Commensurability
3218:Axiomatic system
3166:Angle trisection
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2271:Following Wilson
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2061:
2050:
2044:
2043:
2025:
1913:John Stuart Mill
1885:'s 14th century
1861:
1859:
1858:
1853:
1802:
1800:
1799:
1794:
1792:
1791:
1758:
1757:
1739:
1738:
1708:
1706:
1705:
1700:
1655:
1653:
1652:
1647:
1614:
1612:
1611:
1606:
1546:
1539:
1535:
1532:
1526:
1495:
1487:
1411:
1409:
1408:
1403:
1347:
1345:
1344:
1339:
1300:of the angle at
1292:A textbook proof
1279:
1277:
1276:
1271:
1223:
1221:
1220:
1215:
1123:
1121:
1119:
1118:
1113:
1075:
1073:
1071:
1070:
1065:
1027:
1025:
1023:
1022:
1017:
979:
977:
975:
974:
969:
931:
929:
927:
926:
921:
883:
881:
879:
878:
873:
835:
833:
831:
830:
825:
799:
797:
795:
794:
789:
763:
761:
759:
758:
753:
715:
713:
711:
710:
705:
667:
665:
663:
662:
657:
619:
617:
615:
614:
609:
592:
590:
588:
587:
582:
565:
563:
561:
560:
555:
538:
536:
534:
533:
528:
502:
500:
498:
497:
492:
475:
473:
471:
470:
465:
451:
449:
447:
446:
441:
424:
422:
420:
419:
414:
400:
398:
396:
395:
390:
364:
362:
360:
359:
354:
278:
277:
262:
261:
243:Euler's identity
125:
121:
116:
115:
112:
111:
108:
105:
102:
99:
96:
93:
90:
87:
83:
82:
79:
76:
73:
70:
3866:
3865:
3861:
3860:
3859:
3857:
3856:
3855:
3821:
3820:
3819:
3814:
3803:
3792:
3790:
3779:
3745:Arabian/Islamic
3733:
3722:numeral systems
3611:
3561:
3531:
3479:Heron's formula
3457:
3439:
3331:
3327:Triangle center
3317:Regular polygon
3194:and definitions
3193:
3187:
3149:
3129:
3119:
3081:
3071:
3061:
3051:
3027:
3017:
3000:
2966:
2937:Theon of Smyrna
2582:
2574:
2569:
2519:
2482:
2477:
2468:
2464:
2455:
2451:
2434:
2430:
2419:
2415:
2402:
2398:
2388:
2386:
2377:Kolpas, Sid J.
2375:
2371:
2364:
2334:Dunham, William
2331:
2327:
2296:"Pons Asinorum"
2292:
2288:
2280:A. M. Legendre
2279:
2275:
2270:
2266:
2257:
2253:
2248:
2244:
2239:
2235:
2226:
2222:
2203:
2196:
2187:
2183:
2178:
2174:
2169:
2165:
2149:
2148:
2141:
2119:
2115:
2102:
2098:
2089:
2078:
2069:
2065:
2054:"Pons asinorum"
2052:
2051:
2047:
2040:
2026:
2019:
2015:
1995:
1906:Thomas Campbell
1887:The Philobiblon
1872:
1808:
1805:
1804:
1787:
1783:
1753:
1749:
1734:
1730:
1716:
1713:
1712:
1661:
1658:
1657:
1620:
1617:
1616:
1582:
1579:
1578:
1563:complex numbers
1547:
1536:
1530:
1527:
1512:
1496:
1485:
1373:
1370:
1369:
1324:
1321:
1320:
1286:
1253:
1250:
1249:
1206:
1203:
1202:
1138:Charles Dodgson
1130:
1083:
1080:
1079:
1077:
1035:
1032:
1031:
1029:
987:
984:
983:
981:
939:
936:
935:
933:
891:
888:
887:
885:
843:
840:
839:
837:
807:
804:
803:
801:
771:
768:
767:
765:
723:
720:
719:
717:
675:
672:
671:
669:
627:
624:
623:
621:
600:
597:
596:
594:
573:
570:
569:
567:
546:
543:
542:
540:
510:
507:
506:
504:
483:
480:
479:
477:
459:
456:
455:
453:
432:
429:
428:
426:
408:
405:
404:
402:
372:
369:
368:
366:
339:
336:
335:
333:
319:side-angle-side
260:
255:
175:
123:
119:
84:
67:
63:
17:
12:
11:
5:
3864:
3854:
3853:
3848:
3843:
3838:
3833:
3816:
3815:
3788:
3785:
3784:
3781:
3780:
3778:
3777:
3772:
3767:
3762:
3757:
3752:
3747:
3741:
3739:
3738:Other cultures
3735:
3734:
3732:
3731:
3730:
3729:
3719:
3718:
3717:
3707:
3706:
3705:
3695:
3694:
3693:
3683:
3682:
3681:
3671:
3670:
3669:
3659:
3658:
3657:
3647:
3646:
3645:
3635:
3634:
3633:
3619:
3617:
3613:
3612:
3610:
3609:
3604:
3599:
3594:
3589:
3587:Greek numerals
3584:
3582:Attic numerals
3579:
3573:
3567:
3563:
3562:
3560:
3559:
3554:
3549:
3543:
3541:
3537:
3536:
3533:
3532:
3530:
3529:
3524:
3519:
3514:
3509:
3501:
3496:
3491:
3486:
3481:
3476:
3471:
3465:
3463:
3459:
3458:
3456:
3455:
3449:
3447:
3441:
3440:
3438:
3437:
3432:
3427:
3422:
3417:
3412:
3410:Law of cosines
3407:
3402:
3397:
3392:
3387:
3382:
3377:
3372:
3367:
3362:
3357:
3351:
3349:
3337:
3333:
3332:
3330:
3329:
3324:
3319:
3314:
3309:
3304:
3302:Platonic solid
3299:
3294:
3289:
3284:
3282:Greek numerals
3279:
3274:
3269:
3264:
3259:
3254:
3249:
3248:
3247:
3242:
3232:
3227:
3226:
3225:
3215:
3214:
3213:
3208:
3197:
3195:
3189:
3188:
3186:
3185:
3180:
3179:
3178:
3173:
3168:
3157:
3155:
3151:
3150:
3148:
3147:
3140:
3133:
3123:
3113:
3110:Planisphaerium
3106:
3099:
3092:
3085:
3075:
3065:
3055:
3045:
3038:
3031:
3021:
3011:
3004:
2994:
2987:
2982:
2974:
2972:
2968:
2967:
2965:
2964:
2959:
2954:
2949:
2944:
2939:
2934:
2929:
2924:
2919:
2914:
2909:
2904:
2899:
2894:
2889:
2884:
2879:
2874:
2869:
2864:
2859:
2854:
2849:
2844:
2839:
2834:
2829:
2824:
2819:
2814:
2809:
2804:
2799:
2794:
2789:
2784:
2779:
2774:
2769:
2764:
2759:
2754:
2749:
2744:
2739:
2734:
2729:
2724:
2719:
2714:
2709:
2704:
2699:
2694:
2689:
2684:
2679:
2674:
2669:
2664:
2659:
2654:
2649:
2644:
2639:
2634:
2629:
2624:
2619:
2614:
2609:
2604:
2599:
2594:
2588:
2586:
2580:Mathematicians
2576:
2575:
2568:
2567:
2560:
2553:
2545:
2539:
2538:
2530:
2518:
2517:External links
2515:
2514:
2513:
2509:Vol. 2 (1789)
2496:
2481:
2478:
2476:
2475:
2471:The New Yorker
2462:
2456:A. Battersby,
2449:
2428:
2413:
2396:
2369:
2362:
2325:
2286:
2273:
2264:
2251:
2242:
2233:
2220:
2194:
2181:
2172:
2163:
2139:
2113:
2096:
2076:
2063:
2045:
2039:978-1528770439
2038:
2016:
2014:
2011:
1994:
1991:
1990:
1989:
1978:
1964:ezelsbruggetje
1956:
1927:
1909:
1902:
1891:
1871:
1868:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1790:
1786:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1756:
1752:
1748:
1745:
1742:
1737:
1733:
1729:
1726:
1723:
1720:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1549:
1548:
1499:
1497:
1490:
1484:
1481:
1473:William Dunham
1448:side-side-side
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1337:
1334:
1331:
1328:
1285:
1282:
1269:
1266:
1263:
1260:
1257:
1213:
1210:
1129:
1126:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
967:
964:
961:
958:
955:
952:
949:
946:
943:
919:
916:
913:
910:
907:
904:
901:
898:
895:
871:
868:
865:
862:
859:
856:
853:
850:
847:
823:
820:
817:
814:
811:
787:
784:
781:
778:
775:
751:
748:
745:
742:
739:
736:
733:
730:
727:
703:
700:
697:
694:
691:
688:
685:
682:
679:
655:
652:
649:
646:
643:
640:
637:
634:
631:
607:
604:
580:
577:
553:
550:
526:
523:
520:
517:
514:
490:
487:
463:
439:
436:
412:
388:
385:
382:
379:
376:
352:
349:
346:
343:
296:
295:
274:
273:
270:Proclus' proof
259:
256:
254:
251:
228:Dhū 'l qarnain
174:
171:
163:metaphorically
15:
9:
6:
4:
3:
2:
3863:
3852:
3849:
3847:
3844:
3842:
3839:
3837:
3834:
3832:
3829:
3828:
3826:
3813:
3812:
3807:
3800:
3799:
3786:
3776:
3773:
3771:
3768:
3766:
3763:
3761:
3758:
3756:
3753:
3751:
3748:
3746:
3743:
3742:
3740:
3736:
3728:
3725:
3724:
3723:
3720:
3716:
3713:
3712:
3711:
3708:
3704:
3701:
3700:
3699:
3696:
3692:
3689:
3688:
3687:
3684:
3680:
3677:
3676:
3675:
3672:
3668:
3665:
3664:
3663:
3660:
3656:
3653:
3652:
3651:
3648:
3644:
3641:
3640:
3639:
3636:
3632:
3628:
3627:
3626:
3625:
3621:
3620:
3618:
3614:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3580:
3578:
3575:
3574:
3571:
3568:
3564:
3558:
3555:
3553:
3550:
3548:
3545:
3544:
3542:
3538:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3507:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3466:
3464:
3460:
3454:
3451:
3450:
3448:
3446:
3442:
3436:
3433:
3431:
3428:
3426:
3423:
3421:
3418:
3416:
3415:Pons asinorum
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3385:Hinge theorem
3383:
3381:
3378:
3376:
3373:
3371:
3368:
3366:
3363:
3361:
3358:
3356:
3353:
3352:
3350:
3348:
3347:
3341:
3338:
3334:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3250:
3246:
3243:
3241:
3238:
3237:
3236:
3233:
3231:
3228:
3224:
3221:
3220:
3219:
3216:
3212:
3209:
3207:
3204:
3203:
3202:
3199:
3198:
3196:
3190:
3184:
3181:
3177:
3174:
3172:
3169:
3167:
3164:
3163:
3162:
3159:
3158:
3156:
3152:
3146:
3145:
3141:
3139:
3138:
3134:
3132:
3128:
3124:
3122:
3118:
3114:
3112:
3111:
3107:
3105:
3104:
3100:
3098:
3097:
3093:
3091:
3090:
3086:
3084:
3080:
3076:
3074:
3070:
3066:
3064:
3060:
3056:
3054:
3052:(Aristarchus)
3050:
3046:
3044:
3043:
3039:
3037:
3036:
3032:
3030:
3026:
3022:
3020:
3016:
3012:
3010:
3009:
3005:
3003:
2999:
2995:
2993:
2992:
2988:
2986:
2983:
2981:
2980:
2976:
2975:
2973:
2969:
2963:
2960:
2958:
2957:Zeno of Sidon
2955:
2953:
2950:
2948:
2945:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2898:
2895:
2893:
2890:
2888:
2885:
2883:
2880:
2878:
2875:
2873:
2870:
2868:
2865:
2863:
2860:
2858:
2855:
2853:
2850:
2848:
2845:
2843:
2840:
2838:
2835:
2833:
2830:
2828:
2825:
2823:
2820:
2818:
2815:
2813:
2810:
2808:
2805:
2803:
2800:
2798:
2795:
2793:
2790:
2788:
2785:
2783:
2780:
2778:
2775:
2773:
2770:
2768:
2765:
2763:
2760:
2758:
2755:
2753:
2750:
2748:
2745:
2743:
2740:
2738:
2735:
2733:
2730:
2728:
2725:
2723:
2720:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2685:
2683:
2680:
2678:
2675:
2673:
2670:
2668:
2665:
2663:
2660:
2658:
2655:
2653:
2650:
2648:
2645:
2643:
2640:
2638:
2635:
2633:
2630:
2628:
2625:
2623:
2620:
2618:
2615:
2613:
2610:
2608:
2605:
2603:
2600:
2598:
2595:
2593:
2590:
2589:
2587:
2585:
2581:
2577:
2573:
2566:
2561:
2559:
2554:
2552:
2547:
2546:
2543:
2537:
2536:
2531:
2528:
2524:
2523:Pons asinorum
2521:
2520:
2512:
2508:
2505:
2501:
2497:
2495:
2491:
2488:
2484:
2483:
2472:
2466:
2459:
2453:
2446:
2442:
2438:
2432:
2425:
2422:
2417:
2409:
2408:
2400:
2384:
2380:
2373:
2365:
2363:9780471536567
2359:
2355:
2351:
2347:
2342:
2341:
2335:
2329:
2321:
2317:
2313:
2309:
2305:
2301:
2297:
2290:
2283:
2277:
2268:
2261:
2255:
2246:
2237:
2230:
2224:
2218:
2214:
2210:
2207:
2201:
2199:
2191:
2185:
2176:
2167:
2159:
2153:
2146:
2142:
2140:0-309-08549-7
2136:
2132:
2127:
2126:
2117:
2109:
2108:
2100:
2093:
2087:
2085:
2083:
2081:
2073:
2067:
2059:
2055:
2049:
2041:
2035:
2031:
2024:
2022:
2017:
2010:
2008:
2004:
2003:Marvin Minsky
2000:
1987:
1983:
1979:
1976:
1973:
1969:
1965:
1961:
1957:
1954:
1950:
1946:
1942:
1939:
1935:
1932:
1928:
1926:of economics.
1925:
1924:pons asinorum
1921:
1918:
1914:
1910:
1907:
1903:
1900:
1896:
1895:pons asinorum
1892:
1888:
1884:
1881:
1880:
1879:
1877:
1876:pons asinorum
1867:
1865:
1849:
1846:
1843:
1840:
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1568:
1564:
1560:
1556:
1545:
1542:
1534:
1524:
1520:
1516:
1510:
1509:
1505:
1500:This section
1498:
1494:
1489:
1488:
1480:
1478:
1474:
1470:
1469:
1464:
1460:
1455:
1453:
1449:
1445:
1441:
1438:, but taking
1437:
1433:
1429:
1427:
1423:
1419:
1416:and triangle
1415:
1399:
1396:
1393:
1387:
1384:
1381:
1378:
1367:
1363:
1360: =
1359:
1355:
1351:
1335:
1332:
1329:
1318:
1315: =
1314:
1310:
1305:
1303:
1299:
1290:
1281:
1267:
1261:
1258:
1247:
1243:
1239:
1236: =
1235:
1231:
1228: =
1227:
1211:
1200:
1196:
1192:
1188:
1184:
1180:
1176:
1172:
1168:
1164:
1160:
1156:
1151:
1149:
1145:
1144:
1139:
1135:
1125:
1109:
1106:
1103:
1097:
1094:
1091:
1088:
1061:
1058:
1055:
1049:
1046:
1043:
1040:
1013:
1010:
1007:
1001:
998:
995:
992:
965:
962:
959:
953:
950:
947:
944:
917:
914:
911:
905:
902:
899:
896:
869:
866:
863:
857:
854:
851:
848:
821:
818:
815:
812:
809:
785:
782:
779:
776:
773:
749:
746:
743:
737:
734:
731:
728:
701:
698:
695:
689:
686:
683:
680:
653:
650:
647:
641:
638:
635:
632:
605:
602:
578:
575:
551:
548:
524:
521:
518:
515:
512:
488:
485:
461:
437:
434:
410:
386:
383:
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377:
374:
350:
347:
344:
330:
328:
324:
320:
314:
311:
307:
303:
302:pons asinorum
292:
291:pons asinorum
288:
284:
280:
279:
268:
264:
263:
250:
248:
247:mathematician
244:
240:
236:
234:
233:pons asinorum
230:
229:
224:
220:
219:
213:
211:
207:
203:
199:
195:
190:
188:
184:
180:
179:pons asinorum
170:
168:
164:
161:is also used
160:
159:Pons asinorum
156:
154:
150:
146:
145:
140:
136:
132:
128:
127:
114:
61:
60:
59:pons asinorum
55:
51:
47:
43:
36:
32:
28:
27:pons asinorum
23:
19:
3802:
3789:
3631:Thomas Heath
3622:
3505:
3489:Law of sines
3414:
3345:
3277:Golden ratio
3142:
3135:
3126:
3120:(Theodosius)
3116:
3108:
3101:
3094:
3087:
3078:
3068:
3062:(Hipparchus)
3058:
3048:
3040:
3033:
3024:
3014:
3006:
3001:(Apollonius)
2997:
2989:
2977:
2952:Zeno of Elea
2712:Eratosthenes
2702:Dionysodorus
2534:
2511:Google Books
2506:
2494:Google Books
2489:
2470:
2465:
2457:
2452:
2436:
2431:
2423:
2416:
2406:
2399:
2389:December 22,
2387:. Retrieved
2372:
2339:
2328:
2303:
2299:
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2236:
2228:
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2208:
2205:
2189:
2184:
2175:
2166:
2144:
2124:
2116:
2106:
2099:
2091:
2071:
2066:
2057:
2048:
2029:
2007:Pappus proof
1996:
1985:
1974:
1963:
1945:non sequitur
1940:
1933:
1923:
1894:
1875:
1874:Uses of the
1873:
1863:
1710:
1574:
1570:
1566:
1552:
1537:
1528:
1513:Please help
1501:
1476:
1466:
1456:
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1443:
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1435:
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1413:
1365:
1361:
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1349:
1316:
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1190:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1154:
1152:
1141:
1131:
668:. Therefore
331:
322:
315:
301:
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286:
237:
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226:
216:
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205:
201:
193:
191:
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158:
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142:
134:
58:
57:
39:
34:
31:Oliver Byrne
26:
18:
3698:mathematics
3506:Arithmetica
3103:Ostomachion
3072:(Autolycus)
2991:Arithmetica
2767:Hippocrates
2697:Dinostratus
2682:Dicaearchus
2612:Aristarchus
2487:T. L. Heath
2306:(14): 161.
2070:D.E. Smith
1986:oslí můstek
1975:Eselsbrücke
1920:Law of Rent
1428:are equal.
425:along side
198:Roger Bacon
3825:Categories
3750:Babylonian
3650:arithmetic
3616:History of
3445:Apollonius
3130:(Menelaus)
3089:On Spirals
3008:Catoptrics
2947:Xenocrates
2942:Thymaridas
2927:Theodosius
2912:Theaetetus
2892:Simplicius
2882:Pythagoras
2867:Posidonius
2852:Philonides
2812:Nicomachus
2807:Metrodorus
2797:Menaechmus
2752:Hipparchus
2742:Heliodorus
2692:Diophantus
2677:Democritus
2657:Chrysippus
2627:Archimedes
2622:Apollonius
2592:Anaxagoras
2584:(timeline)
2527:PlanetMath
2480:References
2445:9401587884
1941:åsnebrygga
1934:aasinsilta
1911:Economist
1148:Irish bull
1028:. Finally
3211:Inscribed
2971:Treatises
2962:Zenodorus
2922:Theodorus
2897:Sosigenes
2842:Philolaus
2827:Oenopides
2822:Nicoteles
2817:Nicomedes
2777:Hypsicles
2672:Ctesibius
2662:Cleomedes
2647:Callippus
2632:Autolycus
2617:Aristotle
2597:Anthemius
2504:T. Taylor
2312:2578-4145
2152:cite book
1917:Ricardo's
1899:syllogism
1893:The term
1847:θ
1844:
1838:‖
1832:‖
1829:‖
1823:‖
1814:⋅
1785:‖
1778:‖
1769:⋅
1760:−
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1744:‖
1732:‖
1725:−
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1688:−
1682:‖
1676:‖
1670:−
1664:‖
1641:‖
1635:‖
1629:‖
1623:‖
1557:over the
1502:does not
1391:∠
1376:∠
1327:∠
1265:∠
1256:∠
1209:∠
1101:∠
1098:≅
1086:∠
1053:△
1050:≅
1038:△
1005:∠
1002:≅
990:∠
957:∠
954:≅
942:∠
909:∠
906:≅
894:∠
861:△
858:≅
846:△
816:≅
780:≅
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738:≅
726:∠
693:∠
690:≅
678:∠
645:△
642:≅
630:△
519:≅
381:≅
342:△
327:congruent
289:I.5, the
218:Dulcarnon
215:The name
210:Chaucer's
173:Etymology
48:that the
3775:Japanese
3760:Egyptian
3703:timeline
3691:timeline
3679:timeline
3674:geometry
3667:timeline
3662:calculus
3655:timeline
3643:timeline
3346:Elements
3192:Concepts
3154:Problems
3127:Spherics
3117:Spherics
3082:(Euclid)
3028:(Euclid)
3025:Elements
3018:(Euclid)
2979:Almagest
2887:Serenus
2862:Porphyry
2802:Menelaus
2757:Hippasus
2732:Eutocius
2707:Domninus
2602:Archytas
2535:Elements
2507:Elements
2490:Elements
2439:, 2013,
2383:Archived
2336:(1994).
2320:44764657
1968:mnemonic
1953:causerie
1531:May 2024
1452:Elements
1432:Legendre
1298:bisector
1173:, where
323:Elements
287:Elements
187:Elements
153:triangle
149:converse
144:Elements
42:geometry
35:Elements
3755:Chinese
3710:numbers
3638:algebra
3566:Related
3540:Centers
3336:Results
3206:Central
2877:Ptolemy
2872:Proclus
2837:Perseus
2792:Marinus
2772:Hypatia
2762:Hippias
2737:Geminus
2727:Eudoxus
2717:Eudemus
2687:Diocles
2500:Proclus
2447:, p. 72
2350:Bibcode
2217:3618841
1938:Swedish
1931:Finnish
1915:called
1523:removed
1508:sources
1477:Journal
1122:
1078:
1074:
1030:
1026:
982:
978:
934:
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886:
882:
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798:
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762:
718:
714:
670:
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591:
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541:
537:
505:
501:
478:
474:
454:
450:
427:
423:
403:
399:
367:
363:
334:
310:Proclus
194:Elefuga
122:ass-ih-
46:theorem
3770:Indian
3547:Cyrene
3079:Optics
2998:Conics
2917:Theano
2907:Thales
2902:Sporus
2847:Philon
2832:Pappus
2722:Euclid
2652:Carpus
2642:Bryson
2443:
2360:
2318:
2310:
2215:
2137:
2036:
1972:German
1862:where
1711:Since
1573:, and
1284:Others
1128:Pappus
764:, and
593:, and
253:Proofs
202:elegia
183:bridge
147:. Its
139:Euclid
50:angles
44:, the
3765:Incan
3686:logic
3462:Other
3230:Chord
3223:Axiom
3201:Angle
2857:Plato
2747:Heron
2667:Conon
2316:JSTOR
2213:JSTOR
2013:Notes
1982:Czech
1960:Dutch
1951:- or
1656:then
1311:with
306:proof
131:asses
3727:list
3015:Data
2787:Leon
2637:Bion
2441:ISBN
2391:2021
2358:ISBN
2308:ISSN
2158:link
2135:ISBN
2034:ISBN
1936:and
1929:The
1922:the
1803:and
1615:and
1559:real
1506:any
1504:cite
1424:and
1364:and
1244:and
1232:and
1197:and
1185:and
1169:and
1161:and
932:and
206:fuga
120:PONZ
25:The
3629:by
3343:In
2525:at
2131:202
1980:In
1958:In
1841:cos
1561:or
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1414:BAX
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1140:in
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1358:AB
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1159:AB
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2322:.
2304:3
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2060:.
2042:.
1977:.
1901:.
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1262:=
1259:B
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1199:C
1195:B
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1104:B
1095:D
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1014:B
1011:E
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813:D
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744:A
735:C
732:D
729:A
702:D
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