5137:
108:
6893:
1803:
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1689:
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1815:
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100:
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Suppose S is the common starting point of two rays, and two parallel lines are intersecting those two rays (see figure). Let A, B be the intersections of the first ray with the two parallels, such that B is further away from S than A, and similarly C, D are the intersections of the second ray with
1260:, i.e. it can be used to prove the properties of similar triangles and similar triangles can be used to prove the intercept theorem. By matching identical angles you can always place two similar triangles in one another so that you get the configuration in which the intercept theorem applies; and
5301:.. without trouble or the assistance of any instrument merely set up a stick at the extremity of the shadow cast by the pyramid and, having thus made two triangles by the intercept of the sun's rays, ... showed that the pyramid has to the stick the same ratio which the shadow has to the shadow
1463:
1728:
It took more than 2000 years until all three of them were finally shown to be impossible. This was achieved in the 19th century with the help of algebraic methods, that had become available by then. In order to reformulate the three problems in algebraic terms using
5282:
give a description that strictly speaking does not require the intercept theorem, but can rely on a simple observation only, namely that at a certain point of the day the length of an object's shadow will match its height. Laertius quotes a statement of the philosopher
2282:
1741:). In particular it is important to assure that for two given line segments, a new line segment can be constructed, such that its length equals the product of lengths of the other two. Similarly one needs to be able to construct, for a line segment of length
2292:
The intercept theorem can be used to determine a distance that cannot be measured directly, such as the width of a river or a lake, the height of tall buildings or similar. The graphic to the right illustrates measuring the width of a river. The segments
1380:
3075:
2951:
3895:
2084:
Thales measured the length of the pyramid's base and the height of his pole. Then at the same time of the day he measured the length of the pyramid's shadow and the length of the pole's shadow. This yielded the following data:
904:, then each parallel contains more than one line segment and the ratio of two line segments on one parallel equals the ratio of the according line segments on the other parallel. For instance if there's a third ray starting at
592:
2174:
3448:
3284:
1678:{\displaystyle {\frac {\|\lambda \cdot {\vec {a}}\|}{\|{\vec {a}}\|}}={\frac {\|\lambda \cdot {\vec {b}}\|}{\|{\vec {b}}\|}}={\frac {\|\lambda \cdot ({\vec {a}}+{\vec {b}})\|}{\|{\vec {a}}+{\vec {b}}\|}}=|\lambda |}
1458:
3118:
1230:
For the second equation the converse is true as well, that is if the 3 rays are intercepted by two lines and the ratios of the according line segments on each line are equal, then those 2 lines must be parallel.
3998:
3655:
3554:
4338:
4238:
1225:
1125:
694:
446:
346:
246:
2551:
An elementary proof of the theorem uses triangles of equal area to derive the basic statements about the ratios (claim 1). The other claims then follow by applying the first claim and contradiction.
5289:
Hieronymus says that measured the height of the pyramids by the shadow they cast, taking the observation at the hour when our shadow is of the same length as ourselves (i.e. as our own height).
4810:
5293:
Thales discovered how to obtain the height of pyramids and all other similar objects, namely, by measuring the shadow of the object at the time when a body and its shadow are equal in length.
2825:
2761:
4709:
2482:
2185:
2081:. The following description illustrates the use of the intercept theorem to compute the height of the pyramid. It does not, however, recount Thales' original work, which was lost.
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No original work of Thales has survived. All historical sources that attribute the intercept theorem or related knowledge to him were written centuries after his death.
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If the midpoints of the two non-parallel sides of a trapezoid are connected, then the resulting line segment is parallel to the other two sides of the trapezoid.
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yields V figure with identical measures for which the original theorem now applies. The third statement (converse) however does not remain true for lines.
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holds then the two intercepting lines are parallel. However, the converse of the second statement is not true (see graphic for a counterexample).
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3082:
2543:, who may have used some form of the theorem to determine heights of pyramids in Egypt and to compute the distance of ship from the shore.
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5321:
19:
This article is about the theorem about the ratios of various line segments. For the special case of the inscribed angle theorem, see
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If the midpoints of two triangle sides are connected then the resulting line segment is parallel to the third triangle side (
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The converse of the first statement is true as well, i.e. if the two rays are intercepted by two arbitrary lines and
2277:{\displaystyle D={\frac {C\cdot A}{B}}={\frac {1.63~{\text{m}}\cdot 180~{\text{m}}}{2~{\text{m}}}}=146.7~{\text{m}}}
4622:
2395:
143:
the two parallels such that D is further away from S than C. In this configuration the following statements hold:
6777:
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6343:
6223:
5611:
450:
The ratio of the two segments on the same ray starting at S equals the ratio of the segments on the parallels:
147:
The ratio of any two segments on the first ray equals the ratio of the according segments on the second ray:
6846:
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6491:
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gives an account that may suggest Thales knowing the intercept theorem or at least a special case of it:"
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4531:
2505:
2497:
The intercept theorem can be used to prove that a certain construction yields parallel line (segment)s.
6182:
5998:
2699:
are of equal length. As those triangles share the same baseline, their areas are identical. So we have
1791:. The intercept theorem can be used to show that for both cases, that such a construction is possible.
1375:{\displaystyle \lambda \cdot ({\vec {a}}+{\vec {b}})=\lambda \cdot {\vec {a}}+\lambda \cdot {\vec {b}}}
5973:
5110:
1984:
1952:
1890:
1832:
6717:
6688:
6048:
5903:
3070:{\displaystyle {\frac {|\triangle SCA|}{|\triangle SDA|}}={\frac {|\triangle SCA|}{|\triangle SCB|}}}
2946:{\displaystyle {\frac {|\triangle SCA|}{|\triangle CDA|}}={\frac {|\triangle SCA|}{|\triangle CBA|}}}
2612:
6851:
6585:
6128:
2673:
2644:
1945:
equidistant points, then draw the line through the last point and B and parallel line through the
1704:
There are three famous problems in elementary geometry which were posed by the Greeks in terms of
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The first two statements remain true if the two rays get replaced by two lines intersecting in
5605:
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5087:
4904:
3890:{\displaystyle {\frac {\frac {|SA||SD|}{|SB|}}{|CD|}}={\frac {\frac {|SB||SC|}{|SD|}}{|AB|}}}
2078:
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16:
On ratios of line segments formed when 2 intersecting lines are cut by a pair of parallels
8:
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2016:
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1722:
59:
47:
20:
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2540:
1981:
in the desired ratio. The graphic to the right shows the partition of a line segment
1802:
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1257:
781:
is not located between the two parallels, the original theorem applies directly. If
67:
63:
6772:
6643:
6560:
6316:
6304:
6252:
5988:
5573:
Mit harmonischen Verhältnissen zu
Kegelschnitten: Perlen der klassischen Geometrie.
5115:
Mit harmonischen Verhältnissen zu
Kegelschnitten: Perlen der klassischen Geometrie.
4934:. This is a contradiction, so the assumption could not have been true, which means
1261:
761:, either it lies between the 2 parallels (X figure) or it does not (V figure). If
6413:
6403:
6297:
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5279:
1730:
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6388:
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1244:
55:
4376:
1814:
1688:
587:{\displaystyle {\frac {|SA|}{|SB|}}={\frac {|SC|}{|SD|}}={\frac {|AC|}{|BD|}}}
6911:
6633:
6501:
6471:
6292:
6100:
6043:
5465:
5066:
2179:
Knowing A, B and C he was now able to apply the intercept theorem to compute
1248:
Arranging two similar triangles, so that the intercept theorem can be applied
5627:
2563:
6575:
6363:
6038:
5798:
5788:
5617:
5565:
5527:
5491:
5458:
5359:
Ancient
Mathematics. History of Mathematics in Ancient Greece and Hellenism
5262:
5092:
4035:
1699:
1273:
1264:
the intercept theorem configuration always contains two similar triangles.
51:
5593:
2169:{\displaystyle C=65~{\text{m}}+{\frac {230~{\text{m}}}{2}}=180~{\text{m}}}
6189:
6077:
5783:
5768:
5432:
3443:{\displaystyle {\frac {|SC||AF|}{|SD||AF|}}={\frac {|SA||EC|}{|SB||EC|}}}
3279:{\displaystyle {\frac {|SC||AF|}{|CD||AF|}}={\frac {|SA||EC|}{|AB||EC|}}}
6175:
6094:
6033:
6028:
5968:
5953:
5898:
5883:
5838:
5778:
5763:
5743:
5713:
5678:
5599:
5417:
1453:{\displaystyle \|\lambda {\vec {a}}\|=|\lambda |\cdot \ \|{\vec {a}}\|}
5622:
3113:{\displaystyle {\tfrac {{\text{baseline}}\cdot {\text{altitude}}}{2}}}
99:
5928:
5913:
5863:
5758:
5748:
5733:
5703:
2523:
2511:
713:
705:
78:
74:
6065:
5843:
5688:
5296:
2539:
The theorem is traditionally attributed to the Greek mathematician
2065:
2057:
2041:
6638:
5963:
5958:
5858:
5848:
5823:
43:
5313:, the (translated) original works of Plutarch and Laertius are:
5933:
5808:
2074:
82:
70:
5056:. Dudenverlag, 8. edition, Mannheim 2008, pp. 431–433 (German)
2486:
6309:
6287:
5943:
2492:
2077:
applied the intercept theorem to determine the height of the
2073:
According to some historical sources the Greek mathematician
1277:
3993:{\displaystyle \,{\frac {|SD|}{|CD|}}={\frac {|SB|}{|AB|}}}
3650:{\displaystyle \,{\frac {|SC|}{|SD|}}={\frac {|SA|}{|SB|}}}
3549:{\displaystyle \,{\frac {|SC|}{|CD|}}={\frac {|SA|}{|AB|}}}
1824:
5412:
Lothar Redlin, Ngo Viet, Saleem Watson: "Thales' Shadow",
58:
with a common starting point are intercepted by a pair of
4333:{\displaystyle {\frac {|SA|}{|SB|}}={\frac {|AC|}{|BD|}}}
4233:{\displaystyle {\frac {|SA|}{|SB|}}={\frac {|DG|}{|BD|}}}
1267:
1220:{\displaystyle {\frac {|AE|}{|EC|}}={\frac {|BF|}{|FD|}}}
1120:{\displaystyle {\frac {|AE|}{|BF|}}={\frac {|EC|}{|FD|}}}
689:{\displaystyle {\frac {|SA|}{|AB|}}={\frac {|SC|}{|CD|}}}
441:{\displaystyle {\frac {|SA|}{|SB|}}={\frac {|SC|}{|SD|}}}
341:{\displaystyle {\frac {|SB|}{|AB|}}={\frac {|SD|}{|CD|}}}
241:{\displaystyle {\frac {|SA|}{|AB|}}={\frac {|SC|}{|CD|}}}
1700:
Algebraic formulation of compass and ruler constructions
5571:
Lorenz
Halbeisen, Norbert Hungerbühler, Juan Läuchli:
3087:
741:. In this case there are two scenarios with regard to
4986:
4963:
4940:
4907:
4887:
4867:
4840:
4820:
4719:
4625:
4534:
4501:
4478:
4458:
4435:
4412:
4389:
4347:
4246:
4146:
4094:
4071:
4048:
4005:
3905:
3731:
3698:
3665:
3562:
3461:
3292:
3128:
3085:
2959:
2835:
2769:
2705:
2676:
2647:
2615:
2603:) denote its area and for a line segment its length.
2579:
2398:
2392:
are measured and used to compute the wanted distance
2365:
2332:
2299:
2188:
2111:
2019:
1987:
1955:
1925:
1893:
1867:
1835:
1767:
1747:
1466:
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1133:
1033:
1010:
990:
970:
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910:
890:
870:
847:
827:
807:
801:
lies between the two parallels, then a reflection of
787:
767:
747:
727:
602:
457:
354:
254:
154:
1460:) ensure that the intercept theorem holds. One has
1239:
709:
intercept theorem with a pair of intersecting lines
4992:
4972:
4949:
4926:
4893:
4873:
4853:
4826:
4804:
4703:
4610:
4520:
4487:
4464:
4444:
4421:
4398:
4353:
4332:
4232:
4132:
4080:
4057:
4012:
3992:
3889:
3717:
3684:
3649:
3548:
3442:
3278:
3112:
3069:
2945:
2819:
2755:
2691:
2662:
2633:
2595:
2476:
2384:
2351:
2318:
2276:
2168:
2031:
2005:
1973:
1937:
1911:
1879:
1853:
1783:
1753:
1677:
1452:
1374:
1219:
1119:
1016:
996:
976:
956:
936:
916:
896:
876:
853:
833:
813:
793:
773:
753:
733:
688:
586:
440:
340:
240:
62:. It is equivalent to the theorem about ratios in
5464:
5065:
4805:{\displaystyle |SB_{0}|={\frac {|SD||SA|}{|SC|}}}
2287:
1737:with compass and straightedge constructions (see
6909:
5533:
5230:
2820:{\displaystyle |\triangle SCB|=|\triangle SDA|}
2756:{\displaystyle |\triangle CDA|=|\triangle CBA|}
2052:
5534:Ostermann, Alexander; Wanner, Gerhard (2012).
5231:Ostermann, Alexander; Wanner, Gerhard (2012).
5134:See Agricola/Thomas or the following figure:
5026:(in German). UTB Schöningh. pp. 124–126.
5657:
5643:
4704:{\displaystyle |SB|={\frac {|SD||SA|}{|SC|}}}
2477:{\displaystyle |AB|={\frac {|AC||FE|}{|FC|}}}
700:
94:
5416:, Vol. 73, No. 5 (Dec., 2000), pp. 347-353 (
4429:are not parallel. Then the parallel line to
3079:Plugging in the formula for triangle areas (
1653:
1623:
1618:
1576:
1564:
1549:
1544:
1523:
1511:
1496:
1491:
1470:
1447:
1432:
1407:
1389:
1252:The intercept theorem is closely related to
864:If there are more than two rays starting at
81:, although its first known proof appears in
5105:
5103:
5101:
4712:and on the other hand from claim 1 we have
5650:
5636:
5338:Milestones in Analog and Digital Computing
5178:
2493:Parallel lines in triangles and trapezoids
717:intercept theorem with more than two lines
5497:
5316:Moralia, The Dinner of the Seven Wise Men
4006:
3906:
3563:
3462:
3452:Canceling the common factors results in:
1887:ratio, draw an arbitrary angle in A with
5351:
5330:
5224:
5135:
5098:
5017:
5015:
3899:Using (b) again this simplifies to: (c)
2064:
2056:
1825:Dividing a line segment in a given ratio
1243:
712:
704:
106:
98:
5372:
5268:
5059:
5045:
5043:
2047:
1919:as one leg. On the other leg construct
1024:, then the following equalities holds:
884:or more than two lines intersecting at
6910:
6679:Latin translations of the 12th century
5431:
5406:
5021:
1706:compass and straightedge constructions
1268:Scalar multiplication in vector spaces
6409:Straightedge and compass construction
5631:
5385:
5012:
2534:
1949:th point. This parallel line divides
1256:. It is equivalent to the concept of
6374:Incircle and excircles of a triangle
5395:. Cambridge University Press, 2015,
5205:
5040:
4065:through A. This parallel intersects
1829:To divide an arbitrary line segment
66:. It is traditionally attributed to
5210:(in German). Vieweg. pp. 5–7.
5199:
4611:{\displaystyle |SB|:|SA|=|SD|:|SC|}
1234:
13:
4375:
4034:
3047:
3023:
2992:
2968:
2923:
2899:
2868:
2844:
2800:
2775:
2736:
2711:
2677:
2648:
2573:For a triangle the vertical bars (
2562:
2522:
2510:
924:and intersecting the parallels in
14:
6934:
5587:
5380:An Axiomatic Approach to Geometry
5128:
2095:length of the pyramid base: 230 m
6891:
6878:
5287:(3rd century BC) about Thales: "
2485:
2040:
2006:{\displaystyle {\overline {AB}}}
1974:{\displaystyle {\overline {AB}}}
1912:{\displaystyle {\overline {AB}}}
1854:{\displaystyle {\overline {AB}}}
1813:
1801:
1687:
1240:Similarity and similar triangles
4042:Draw an additional parallel to
1761:, a new line segment of length
1694:
6711:A History of Greek Mathematics
6224:The Quadrature of the Parabola
5472:. AMS. pp. 10–13, 16–18.
5439:. BLoomsbury. pp. 84–87.
5437:Teaching and Learning Geometry
5172:
5073:. AMS. pp. 10–13, 16–18.
4881:and have the same distance to
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3611:
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3510:
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3399:
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3331:
3324:
3313:
3308:
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3269:
3258:
3253:
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3219:
3208:
3194:
3183:
3178:
3167:
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3149:
3144:
3133:
3060:
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3019:
3005:
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2895:
2881:
2864:
2857:
2840:
2813:
2796:
2788:
2771:
2749:
2732:
2724:
2707:
2634:{\displaystyle CA\parallel BD}
2589:
2581:
2467:
2456:
2449:
2438:
2433:
2422:
2411:
2400:
2378:
2367:
2345:
2334:
2312:
2301:
2288:Measuring the width of a river
2089:height of the pole (A): 1.63 m
1671:
1663:
1647:
1632:
1615:
1609:
1594:
1585:
1558:
1538:
1505:
1485:
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1414:
1401:
1366:
1345:
1327:
1321:
1306:
1297:
1210:
1199:
1192:
1181:
1167:
1156:
1149:
1138:
1110:
1099:
1092:
1081:
1067:
1056:
1049:
1038:
679:
668:
661:
650:
636:
625:
618:
607:
577:
566:
559:
548:
534:
523:
516:
505:
491:
480:
473:
462:
431:
420:
413:
402:
388:
377:
370:
359:
331:
320:
313:
302:
288:
277:
270:
259:
231:
220:
213:
202:
188:
177:
170:
159:
73:. It was known to the ancient
1:
5623:intercept theorem interactive
5425:
5323:Lives of Eminent Philosophers
2692:{\displaystyle \triangle CBA}
2663:{\displaystyle \triangle CDA}
2506:Midpoint theorem of triangles
36:basic proportionality theorem
6492:Intersecting secants theorem
5501:The Four Pillars of Geometry
5468:; Friedrich, Thomas (2008).
5325:, Chapter 1. Thales, para.27
5069:; Friedrich, Thomas (2008).
2053:Height of the Cheops pyramid
1998:
1966:
1904:
1846:
50:about the ratios of various
7:
6487:Intersecting chords theorem
6354:Doctrine of proportionality
5583:, pp. 191–208 (German)
5393:Computation, Proof, Machine
5382:. Springer, 2013, pp. 10–13
5159:does not necessarily imply
5153:| / |
5145:| / |
5125:, pp. 191–208 (German)
4521:{\displaystyle B_{0}\neq B}
2098:shadow of the pyramid: 65 m
2092:shadow of the pole (B): 2 m
130:does not necessarily imply
124:| / |
116:| / |
103:intercept theorem with rays
10:
6939:
6923:Theorems in plane geometry
6183:On the Sphere and Cylinder
6136:On the Sizes and Distances
5054:SchĂĽlerduden: Mathematik I
4374:
4367:
4033:
4026:
4013:{\displaystyle \,\square }
2561:
2554:
2518:
2502:
1811:Construction of an inverse
1808:
1796:
701:Extensions and conclusions
95:Formulation of the theorem
18:
6885:Ancient Greece portal
6874:
6824:
6702:
6689:Philosophy of mathematics
6659:
6652:
6626:
6604:Ptolemy's table of chords
6548:
6530:
6429:
6422:
6278:
6240:
6057:
5665:
5659:Ancient Greek mathematics
4133:{\displaystyle |AC|=|DG|}
2596:{\displaystyle |\ldots |}
1799:Construction of a product
6556:Aristarchus's inequality
6129:On Conoids and Spheroids
5498:Stillwell, John (2005).
5006:
4993:{\displaystyle \square }
4861:are on the same side of
4354:{\displaystyle \square }
2546:
54:that are created if two
6664:Ancient Greek astronomy
6477:Inscribed angle theorem
6467:Greek geometric algebra
6122:Measurement of a Circle
5538:Geometry by Its History
5235:Geometry by Its History
5180:Kazarinoff, Nicholas D.
4927:{\displaystyle B=B_{0}}
3659:Now use (b) to replace
3120:) transforms that into
6898:Mathematics portal
6684:Non-Euclidean geometry
6639:Mouseion of Alexandria
6512:Tangent-secant theorem
6462:Geometric mean theorem
6447:Exterior angle theorem
6442:Angle bisector theorem
6146:On Sizes and Distances
5168:
4994:
4974:
4951:
4928:
4895:
4875:
4855:
4828:
4806:
4705:
4612:
4522:
4489:
4466:
4446:
4423:
4400:
4380:
4355:
4334:
4234:
4134:
4082:
4059:
4039:
4014:
3994:
3891:
3719:
3686:
3651:
3550:
3444:
3280:
3114:
3071:
2947:
2821:
2757:
2693:
2664:
2635:
2597:
2567:
2527:
2515:
2478:
2386:
2353:
2320:
2278:
2170:
2102:From this he computed
2070:
2062:
2033:
2007:
1975:
1939:
1913:
1881:
1855:
1785:
1784:{\displaystyle a^{-1}}
1755:
1679:
1454:
1376:
1249:
1221:
1121:
1018:
998:
984:is further away from
978:
958:
938:
918:
898:
878:
855:
835:
815:
795:
775:
755:
735:
718:
710:
690:
588:
442:
342:
242:
139:
104:
6586:Pappus's area theorem
6522:Theorem of the gnomon
6399:Quadratrix of Hippias
6322:Circles of Apollonius
6270:Problem of Apollonius
6248:Constructible numbers
6072:Archimedes Palimpsest
5604:Alexander Bogomolny:
5542:. Springer. pp.
5239:. Springer. pp.
5139:
4995:
4975:
4952:
4929:
4896:
4876:
4856:
4854:{\displaystyle B_{0}}
4829:
4807:
4706:
4613:
4523:
4490:
4467:
4447:
4424:
4401:
4379:
4356:
4335:
4235:
4135:
4083:
4060:
4038:
4015:
3995:
3892:
3720:
3687:
3652:
3551:
3445:
3281:
3115:
3072:
2948:
2827:as well. This yields
2822:
2758:
2694:
2665:
2636:
2598:
2566:
2526:
2514:
2479:
2387:
2354:
2321:
2279:
2171:
2068:
2060:
2034:
2008:
1976:
1940:
1914:
1882:
1856:
1786:
1756:
1733:, one needs to match
1680:
1455:
1377:
1282:scalar multiplication
1247:
1222:
1122:
1019:
999:
979:
959:
939:
919:
899:
879:
856:
836:
816:
796:
776:
756:
736:
716:
708:
691:
589:
443:
343:
243:
110:
102:
40:side splitter theorem
6802:prehistoric counting
6599:Ptolemy's inequality
6540:Apollonius's theorem
6379:Method of exhaustion
6349:Diophantine equation
6339:Circumscribed circle
6156:On the Moving Sphere
5504:. Springer. p.
5414:Mathematics Magazine
5206:Kunz, Ernst (1991).
5186:, Dover, p. 3,
5111:Norbert HungerbĂĽhler
4984:
4980:are indeed parallel
4961:
4938:
4905:
4885:
4865:
4838:
4818:
4717:
4623:
4532:
4499:
4476:
4456:
4433:
4410:
4387:
4345:
4244:
4144:
4092:
4069:
4046:
4003:
3903:
3729:
3718:{\displaystyle |SC|}
3696:
3685:{\displaystyle |SA|}
3663:
3560:
3459:
3290:
3126:
3083:
2957:
2833:
2767:
2703:
2674:
2645:
2613:
2577:
2396:
2385:{\displaystyle |FE|}
2363:
2352:{\displaystyle |CA|}
2330:
2319:{\displaystyle |CF|}
2297:
2186:
2109:
2048:Measuring and survey
2017:
1985:
1953:
1923:
1891:
1865:
1833:
1765:
1745:
1739:constructible number
1713:Trisecting the angle
1464:
1386:
1288:
1131:
1031:
1008:
988:
968:
948:
928:
908:
888:
868:
845:
825:
805:
785:
765:
745:
725:
600:
455:
352:
252:
152:
6888: •
6694:Neusis construction
6614:Spiral of Theodorus
6507:Pythagorean theorem
6452:Euclidean algorithm
6394:Lune of Hippocrates
6263:Squaring the circle
6019:Theon of Alexandria
5694:Aristaeus the Elder
5470:Elementary Geometry
5184:Ruler and the Round
5071:Elementary Geometry
5022:Schupp, H. (1977).
4140:and due to claim 1
4088:in G. Then one has
2641:, the altitudes of
2032:{\displaystyle 5:3}
1938:{\displaystyle m+n}
1880:{\displaystyle m:n}
1723:Squaring the circle
68:Greek mathematician
48:elementary geometry
6918:Euclidean geometry
6581:Menelaus's theorem
6571:Irrational numbers
6384:Parallel postulate
6359:Euclidean geometry
6327:Apollonian circles
5869:Isidore of Miletus
5610:and in particular
5357:Dietmar Herrmann:
5340:. Springer, 2021,
5336:Herbert Bruderer:
5291:". Pliny writes: "
5169:
5109:Lorenz Halbeisen,
5024:Elementargeometrie
4990:
4973:{\displaystyle BD}
4970:
4950:{\displaystyle AC}
4947:
4924:
4891:
4871:
4851:
4824:
4802:
4701:
4608:
4518:
4488:{\displaystyle SA}
4485:
4462:
4445:{\displaystyle AC}
4442:
4422:{\displaystyle BD}
4419:
4399:{\displaystyle AC}
4396:
4381:
4351:
4330:
4230:
4130:
4081:{\displaystyle BD}
4078:
4058:{\displaystyle SD}
4055:
4040:
4010:
3990:
3887:
3715:
3682:
3647:
3546:
3440:
3276:
3110:
3108:
3067:
2943:
2817:
2753:
2689:
2660:
2631:
2593:
2568:
2535:Historical aspects
2528:
2516:
2474:
2382:
2349:
2316:
2274:
2166:
2071:
2063:
2029:
2003:
1971:
1935:
1909:
1877:
1851:
1781:
1751:
1675:
1450:
1372:
1250:
1217:
1117:
1014:
994:
974:
954:
934:
914:
894:
874:
851:
831:
811:
791:
771:
751:
731:
719:
711:
686:
584:
438:
338:
238:
140:
105:
42:, is an important
6905:
6904:
6870:
6869:
6622:
6621:
6609:Ptolemy's theorem
6482:Intercept theorem
6332:Apollonian gasket
6258:Doubling the cube
6231:The Sand Reckoner
5595:Intercept Theorem
5553:978-3-642-29163-0
5515:978-0-387-25530-9
5378:Francis Borceux:
5367:978-3-662-66493-3
5361:, Springer 2022,
5276:Diogenes Laertius
5250:978-3-642-29163-0
5004:
5003:
4894:{\displaystyle S}
4874:{\displaystyle S}
4827:{\displaystyle B}
4800:
4699:
4618:is true, we have
4465:{\displaystyle D}
4365:
4364:
4328:
4285:
4228:
4185:
4024:
4023:
3988:
3945:
3885:
3866:
3806:
3787:
3645:
3602:
3544:
3501:
3438:
3363:
3274:
3199:
3107:
3101:
3093:
3065:
3010:
2941:
2886:
2541:Thales of Miletus
2532:
2531:
2472:
2272:
2268:
2258:
2255:
2251:
2242:
2238:
2228:
2224:
2211:
2164:
2160:
2150:
2144:
2140:
2127:
2123:
2069:computing C and D
2001:
1969:
1907:
1849:
1822:
1821:
1754:{\displaystyle a}
1718:Doubling the cube
1657:
1650:
1635:
1612:
1597:
1568:
1561:
1541:
1515:
1508:
1488:
1444:
1431:
1404:
1369:
1348:
1324:
1309:
1258:similar triangles
1215:
1172:
1115:
1072:
1017:{\displaystyle E}
997:{\displaystyle S}
977:{\displaystyle F}
957:{\displaystyle F}
937:{\displaystyle E}
917:{\displaystyle S}
897:{\displaystyle S}
877:{\displaystyle S}
854:{\displaystyle S}
834:{\displaystyle C}
814:{\displaystyle A}
794:{\displaystyle S}
774:{\displaystyle S}
754:{\displaystyle S}
734:{\displaystyle S}
684:
641:
582:
539:
496:
436:
393:
336:
293:
236:
193:
64:similar triangles
28:intercept theorem
6930:
6896:
6895:
6883:
6882:
6881:
6657:
6656:
6644:Platonic Academy
6591:Problem II.8 of
6561:Crossbar theorem
6517:Thales's theorem
6457:Euclid's theorem
6427:
6426:
6344:Commensurability
6305:Axiomatic system
6253:Angle trisection
6218:
6208:
6170:
6160:
6150:
6140:
6116:
6106:
6089:
5652:
5645:
5638:
5629:
5628:
5607:Thales' Theorems
5557:
5541:
5519:
5483:
5450:
5420:
5410:
5404:
5389:
5383:
5376:
5370:
5355:
5349:
5334:
5328:
5306:Thales biography
5272:
5266:
5254:
5238:
5228:
5222:
5221:
5203:
5197:
5196:
5176:
5170:
5166:
5162:
5158:
5132:
5126:
5113:, Juan Läuchli:
5107:
5096:
5084:
5063:
5057:
5047:
5038:
5037:
5019:
4999:
4997:
4996:
4991:
4979:
4977:
4976:
4971:
4956:
4954:
4953:
4948:
4933:
4931:
4930:
4925:
4923:
4922:
4900:
4898:
4897:
4892:
4880:
4878:
4877:
4872:
4860:
4858:
4857:
4852:
4850:
4849:
4833:
4831:
4830:
4825:
4811:
4809:
4808:
4803:
4801:
4799:
4798:
4787:
4781:
4780:
4769:
4764:
4753:
4747:
4742:
4737:
4736:
4724:
4710:
4708:
4707:
4702:
4700:
4698:
4697:
4686:
4680:
4679:
4668:
4663:
4652:
4646:
4641:
4630:
4617:
4615:
4614:
4609:
4607:
4596:
4588:
4577:
4569:
4558:
4550:
4539:
4527:
4525:
4524:
4519:
4511:
4510:
4494:
4492:
4491:
4486:
4471:
4469:
4468:
4463:
4451:
4449:
4448:
4443:
4428:
4426:
4425:
4420:
4405:
4403:
4402:
4397:
4372:
4371:
4360:
4358:
4357:
4352:
4339:
4337:
4336:
4331:
4329:
4327:
4326:
4315:
4309:
4308:
4297:
4291:
4286:
4284:
4283:
4272:
4266:
4265:
4254:
4248:
4239:
4237:
4236:
4231:
4229:
4227:
4226:
4215:
4209:
4208:
4197:
4191:
4186:
4184:
4183:
4172:
4166:
4165:
4154:
4148:
4139:
4137:
4136:
4131:
4129:
4118:
4110:
4099:
4087:
4085:
4084:
4079:
4064:
4062:
4061:
4056:
4031:
4030:
4019:
4017:
4016:
4011:
3999:
3997:
3996:
3991:
3989:
3987:
3986:
3975:
3969:
3968:
3957:
3951:
3946:
3944:
3943:
3932:
3926:
3925:
3914:
3908:
3896:
3894:
3893:
3888:
3886:
3884:
3883:
3872:
3865:
3864:
3853:
3847:
3846:
3835:
3830:
3819:
3813:
3812:
3807:
3805:
3804:
3793:
3786:
3785:
3774:
3768:
3767:
3756:
3751:
3740:
3734:
3733:
3724:
3722:
3721:
3716:
3714:
3703:
3691:
3689:
3688:
3683:
3681:
3670:
3656:
3654:
3653:
3648:
3646:
3644:
3643:
3632:
3626:
3625:
3614:
3608:
3603:
3601:
3600:
3589:
3583:
3582:
3571:
3565:
3555:
3553:
3552:
3547:
3545:
3543:
3542:
3531:
3525:
3524:
3513:
3507:
3502:
3500:
3499:
3488:
3482:
3481:
3470:
3464:
3449:
3447:
3446:
3441:
3439:
3437:
3436:
3425:
3420:
3409:
3403:
3402:
3391:
3386:
3375:
3369:
3364:
3362:
3361:
3350:
3345:
3334:
3328:
3327:
3316:
3311:
3300:
3294:
3285:
3283:
3282:
3277:
3275:
3273:
3272:
3261:
3256:
3245:
3239:
3238:
3227:
3222:
3211:
3205:
3200:
3198:
3197:
3186:
3181:
3170:
3164:
3163:
3152:
3147:
3136:
3130:
3119:
3117:
3116:
3111:
3109:
3103:
3102:
3099:
3094:
3091:
3088:
3076:
3074:
3073:
3068:
3066:
3064:
3063:
3046:
3040:
3039:
3022:
3016:
3011:
3009:
3008:
2991:
2985:
2984:
2967:
2961:
2952:
2950:
2949:
2944:
2942:
2940:
2939:
2922:
2916:
2915:
2898:
2892:
2887:
2885:
2884:
2867:
2861:
2860:
2843:
2837:
2826:
2824:
2823:
2818:
2816:
2799:
2791:
2774:
2762:
2760:
2759:
2754:
2752:
2735:
2727:
2710:
2698:
2696:
2695:
2690:
2669:
2667:
2666:
2661:
2640:
2638:
2637:
2632:
2602:
2600:
2599:
2594:
2592:
2584:
2559:
2558:
2500:
2499:
2489:
2483:
2481:
2480:
2475:
2473:
2471:
2470:
2459:
2453:
2452:
2441:
2436:
2425:
2419:
2414:
2403:
2391:
2389:
2388:
2383:
2381:
2370:
2358:
2356:
2355:
2350:
2348:
2337:
2325:
2323:
2322:
2317:
2315:
2304:
2283:
2281:
2280:
2275:
2273:
2270:
2266:
2259:
2257:
2256:
2253:
2249:
2244:
2243:
2240:
2236:
2229:
2226:
2222:
2217:
2212:
2207:
2196:
2175:
2173:
2172:
2167:
2165:
2162:
2158:
2151:
2146:
2145:
2142:
2138:
2133:
2128:
2125:
2121:
2061:measuring pieces
2044:
2038:
2036:
2035:
2030:
2012:
2010:
2009:
2004:
2002:
1997:
1989:
1980:
1978:
1977:
1972:
1970:
1965:
1957:
1944:
1942:
1941:
1936:
1918:
1916:
1915:
1910:
1908:
1903:
1895:
1886:
1884:
1883:
1878:
1860:
1858:
1857:
1852:
1850:
1845:
1837:
1817:
1805:
1794:
1793:
1790:
1788:
1787:
1782:
1780:
1779:
1760:
1758:
1757:
1752:
1735:field operations
1731:field extensions
1691:
1684:
1682:
1681:
1676:
1674:
1666:
1658:
1656:
1652:
1651:
1643:
1637:
1636:
1628:
1621:
1614:
1613:
1605:
1599:
1598:
1590:
1574:
1569:
1567:
1563:
1562:
1554:
1547:
1543:
1542:
1534:
1521:
1516:
1514:
1510:
1509:
1501:
1494:
1490:
1489:
1481:
1468:
1459:
1457:
1456:
1451:
1446:
1445:
1437:
1429:
1425:
1417:
1406:
1405:
1397:
1381:
1379:
1378:
1373:
1371:
1370:
1362:
1350:
1349:
1341:
1326:
1325:
1317:
1311:
1310:
1302:
1235:Related concepts
1226:
1224:
1223:
1218:
1216:
1214:
1213:
1202:
1196:
1195:
1184:
1178:
1173:
1171:
1170:
1159:
1153:
1152:
1141:
1135:
1126:
1124:
1123:
1118:
1116:
1114:
1113:
1102:
1096:
1095:
1084:
1078:
1073:
1071:
1070:
1059:
1053:
1052:
1041:
1035:
1023:
1021:
1020:
1015:
1003:
1001:
1000:
995:
983:
981:
980:
975:
963:
961:
960:
955:
943:
941:
940:
935:
923:
921:
920:
915:
903:
901:
900:
895:
883:
881:
880:
875:
860:
858:
857:
852:
840:
838:
837:
832:
820:
818:
817:
812:
800:
798:
797:
792:
780:
778:
777:
772:
760:
758:
757:
752:
740:
738:
737:
732:
695:
693:
692:
687:
685:
683:
682:
671:
665:
664:
653:
647:
642:
640:
639:
628:
622:
621:
610:
604:
593:
591:
590:
585:
583:
581:
580:
569:
563:
562:
551:
545:
540:
538:
537:
526:
520:
519:
508:
502:
497:
495:
494:
483:
477:
476:
465:
459:
447:
445:
444:
439:
437:
435:
434:
423:
417:
416:
405:
399:
394:
392:
391:
380:
374:
373:
362:
356:
347:
345:
344:
339:
337:
335:
334:
323:
317:
316:
305:
299:
294:
292:
291:
280:
274:
273:
262:
256:
247:
245:
244:
239:
237:
235:
234:
223:
217:
216:
205:
199:
194:
192:
191:
180:
174:
173:
162:
156:
137:
133:
129:
32:Thales's theorem
30:, also known as
21:Thales's theorem
6938:
6937:
6933:
6932:
6931:
6929:
6928:
6927:
6908:
6907:
6906:
6901:
6890:
6879:
6877:
6866:
6832:Arabian/Islamic
6820:
6809:numeral systems
6698:
6648:
6618:
6566:Heron's formula
6544:
6526:
6418:
6414:Triangle center
6404:Regular polygon
6281:and definitions
6280:
6274:
6236:
6216:
6206:
6168:
6158:
6148:
6138:
6114:
6104:
6087:
6053:
6024:Theon of Smyrna
5669:
5661:
5656:
5613:Thales' Theorem
5590:
5575:Springer 2016,
5554:
5516:
5480:
5447:
5428:
5423:
5411:
5407:
5390:
5386:
5377:
5373:
5356:
5352:
5335:
5331:
5273:
5269:
5251:
5229:
5225:
5218:
5204:
5200:
5194:
5177:
5173:
5164:
5163:is parallel to
5160:
5149:| = |
5140:
5133:
5129:
5117:Springer 2016,
5108:
5099:
5081:
5064:
5060:
5048:
5041:
5034:
5020:
5013:
5009:
4985:
4982:
4981:
4962:
4959:
4958:
4939:
4936:
4935:
4918:
4914:
4906:
4903:
4902:
4886:
4883:
4882:
4866:
4863:
4862:
4845:
4841:
4839:
4836:
4835:
4819:
4816:
4815:
4813:
4794:
4783:
4782:
4776:
4765:
4760:
4749:
4748:
4746:
4738:
4732:
4728:
4720:
4718:
4715:
4714:
4713:
4711:
4693:
4682:
4681:
4675:
4664:
4659:
4648:
4647:
4645:
4637:
4626:
4624:
4621:
4620:
4619:
4603:
4592:
4584:
4573:
4565:
4554:
4546:
4535:
4533:
4530:
4529:
4506:
4502:
4500:
4497:
4496:
4477:
4474:
4473:
4457:
4454:
4453:
4434:
4431:
4430:
4411:
4408:
4407:
4388:
4385:
4384:
4370:
4346:
4343:
4342:
4322:
4311:
4310:
4304:
4293:
4292:
4290:
4279:
4268:
4267:
4261:
4250:
4249:
4247:
4245:
4242:
4241:
4222:
4211:
4210:
4204:
4193:
4192:
4190:
4179:
4168:
4167:
4161:
4150:
4149:
4147:
4145:
4142:
4141:
4125:
4114:
4106:
4095:
4093:
4090:
4089:
4070:
4067:
4066:
4047:
4044:
4043:
4029:
4004:
4001:
4000:
3982:
3971:
3970:
3964:
3953:
3952:
3950:
3939:
3928:
3927:
3921:
3910:
3909:
3907:
3904:
3901:
3900:
3879:
3868:
3867:
3860:
3849:
3848:
3842:
3831:
3826:
3815:
3814:
3811:
3800:
3789:
3788:
3781:
3770:
3769:
3763:
3752:
3747:
3736:
3735:
3732:
3730:
3727:
3726:
3710:
3699:
3697:
3694:
3693:
3677:
3666:
3664:
3661:
3660:
3639:
3628:
3627:
3621:
3610:
3609:
3607:
3596:
3585:
3584:
3578:
3567:
3566:
3564:
3561:
3558:
3557:
3538:
3527:
3526:
3520:
3509:
3508:
3506:
3495:
3484:
3483:
3477:
3466:
3465:
3463:
3460:
3457:
3456:
3432:
3421:
3416:
3405:
3404:
3398:
3387:
3382:
3371:
3370:
3368:
3357:
3346:
3341:
3330:
3329:
3323:
3312:
3307:
3296:
3295:
3293:
3291:
3288:
3287:
3268:
3257:
3252:
3241:
3240:
3234:
3223:
3218:
3207:
3206:
3204:
3193:
3182:
3177:
3166:
3165:
3159:
3148:
3143:
3132:
3131:
3129:
3127:
3124:
3123:
3098:
3090:
3089:
3086:
3084:
3081:
3080:
3059:
3042:
3041:
3035:
3018:
3017:
3015:
3004:
2987:
2986:
2980:
2963:
2962:
2960:
2958:
2955:
2954:
2935:
2918:
2917:
2911:
2894:
2893:
2891:
2880:
2863:
2862:
2856:
2839:
2838:
2836:
2834:
2831:
2830:
2812:
2795:
2787:
2770:
2768:
2765:
2764:
2748:
2731:
2723:
2706:
2704:
2701:
2700:
2675:
2672:
2671:
2646:
2643:
2642:
2614:
2611:
2610:
2588:
2580:
2578:
2575:
2574:
2557:
2549:
2537:
2495:
2466:
2455:
2454:
2448:
2437:
2432:
2421:
2420:
2418:
2410:
2399:
2397:
2394:
2393:
2377:
2366:
2364:
2361:
2360:
2344:
2333:
2331:
2328:
2327:
2311:
2300:
2298:
2295:
2294:
2290:
2269:
2252:
2245:
2239:
2225:
2218:
2216:
2197:
2195:
2187:
2184:
2183:
2161:
2141:
2134:
2132:
2124:
2110:
2107:
2106:
2079:Cheops' pyramid
2055:
2050:
2018:
2015:
2014:
1990:
1988:
1986:
1983:
1982:
1958:
1956:
1954:
1951:
1950:
1924:
1921:
1920:
1896:
1894:
1892:
1889:
1888:
1866:
1863:
1862:
1838:
1836:
1834:
1831:
1830:
1827:
1772:
1768:
1766:
1763:
1762:
1746:
1743:
1742:
1702:
1697:
1670:
1662:
1642:
1641:
1627:
1626:
1622:
1604:
1603:
1589:
1588:
1575:
1573:
1553:
1552:
1548:
1533:
1532:
1522:
1520:
1500:
1499:
1495:
1480:
1479:
1469:
1467:
1465:
1462:
1461:
1436:
1435:
1421:
1413:
1396:
1395:
1387:
1384:
1383:
1361:
1360:
1340:
1339:
1316:
1315:
1301:
1300:
1289:
1286:
1285:
1284:(in particular
1280:concerning the
1270:
1242:
1237:
1209:
1198:
1197:
1191:
1180:
1179:
1177:
1166:
1155:
1154:
1148:
1137:
1136:
1134:
1132:
1129:
1128:
1109:
1098:
1097:
1091:
1080:
1079:
1077:
1066:
1055:
1054:
1048:
1037:
1036:
1034:
1032:
1029:
1028:
1009:
1006:
1005:
989:
986:
985:
969:
966:
965:
949:
946:
945:
929:
926:
925:
909:
906:
905:
889:
886:
885:
869:
866:
865:
846:
843:
842:
826:
823:
822:
806:
803:
802:
786:
783:
782:
766:
763:
762:
746:
743:
742:
726:
723:
722:
703:
678:
667:
666:
660:
649:
648:
646:
635:
624:
623:
617:
606:
605:
603:
601:
598:
597:
576:
565:
564:
558:
547:
546:
544:
533:
522:
521:
515:
504:
503:
501:
490:
479:
478:
472:
461:
460:
458:
456:
453:
452:
451:
430:
419:
418:
412:
401:
400:
398:
387:
376:
375:
369:
358:
357:
355:
353:
350:
349:
330:
319:
318:
312:
301:
300:
298:
287:
276:
275:
269:
258:
257:
255:
253:
250:
249:
230:
219:
218:
212:
201:
200:
198:
187:
176:
175:
169:
158:
157:
155:
153:
150:
149:
148:
135:
134:is parallel to
131:
120:| = |
111:
97:
24:
17:
12:
11:
5:
6936:
6926:
6925:
6920:
6903:
6902:
6875:
6872:
6871:
6868:
6867:
6865:
6864:
6859:
6854:
6849:
6844:
6839:
6834:
6828:
6826:
6825:Other cultures
6822:
6821:
6819:
6818:
6817:
6816:
6806:
6805:
6804:
6794:
6793:
6792:
6782:
6781:
6780:
6770:
6769:
6768:
6758:
6757:
6756:
6746:
6745:
6744:
6734:
6733:
6732:
6722:
6721:
6720:
6706:
6704:
6700:
6699:
6697:
6696:
6691:
6686:
6681:
6676:
6674:Greek numerals
6671:
6669:Attic numerals
6666:
6660:
6654:
6650:
6649:
6647:
6646:
6641:
6636:
6630:
6628:
6624:
6623:
6620:
6619:
6617:
6616:
6611:
6606:
6601:
6596:
6588:
6583:
6578:
6573:
6568:
6563:
6558:
6552:
6550:
6546:
6545:
6543:
6542:
6536:
6534:
6528:
6527:
6525:
6524:
6519:
6514:
6509:
6504:
6499:
6497:Law of cosines
6494:
6489:
6484:
6479:
6474:
6469:
6464:
6459:
6454:
6449:
6444:
6438:
6436:
6424:
6420:
6419:
6417:
6416:
6411:
6406:
6401:
6396:
6391:
6389:Platonic solid
6386:
6381:
6376:
6371:
6369:Greek numerals
6366:
6361:
6356:
6351:
6346:
6341:
6336:
6335:
6334:
6329:
6319:
6314:
6313:
6312:
6302:
6301:
6300:
6295:
6284:
6282:
6276:
6275:
6273:
6272:
6267:
6266:
6265:
6260:
6255:
6244:
6242:
6238:
6237:
6235:
6234:
6227:
6220:
6210:
6200:
6197:Planisphaerium
6193:
6186:
6179:
6172:
6162:
6152:
6142:
6132:
6125:
6118:
6108:
6098:
6091:
6081:
6074:
6069:
6061:
6059:
6055:
6054:
6052:
6051:
6046:
6041:
6036:
6031:
6026:
6021:
6016:
6011:
6006:
6001:
5996:
5991:
5986:
5981:
5976:
5971:
5966:
5961:
5956:
5951:
5946:
5941:
5936:
5931:
5926:
5921:
5916:
5911:
5906:
5901:
5896:
5891:
5886:
5881:
5876:
5871:
5866:
5861:
5856:
5851:
5846:
5841:
5836:
5831:
5826:
5821:
5816:
5811:
5806:
5801:
5796:
5791:
5786:
5781:
5776:
5771:
5766:
5761:
5756:
5751:
5746:
5741:
5736:
5731:
5726:
5721:
5716:
5711:
5706:
5701:
5696:
5691:
5686:
5681:
5675:
5673:
5667:Mathematicians
5663:
5662:
5655:
5654:
5647:
5640:
5632:
5626:
5625:
5620:
5602:
5589:
5588:External links
5586:
5585:
5584:
5569:
5552:
5531:
5514:
5495:
5478:
5466:Agricola, Ilka
5462:
5445:
5427:
5424:
5422:
5421:
5405:
5391:Gilles Dowek:
5384:
5371:
5350:
5329:
5267:
5249:
5223:
5216:
5198:
5192:
5171:
5127:
5097:
5079:
5067:Agricola, Ilka
5058:
5039:
5032:
5010:
5008:
5005:
5002:
5001:
4989:
4969:
4966:
4946:
4943:
4921:
4917:
4913:
4910:
4901:, which means
4890:
4870:
4848:
4844:
4823:
4797:
4793:
4790:
4786:
4779:
4775:
4772:
4768:
4763:
4759:
4756:
4752:
4745:
4741:
4735:
4731:
4727:
4723:
4696:
4692:
4689:
4685:
4678:
4674:
4671:
4667:
4662:
4658:
4655:
4651:
4644:
4640:
4636:
4633:
4629:
4606:
4602:
4599:
4595:
4591:
4587:
4583:
4580:
4576:
4572:
4568:
4564:
4561:
4557:
4553:
4549:
4545:
4542:
4538:
4517:
4514:
4509:
4505:
4484:
4481:
4461:
4441:
4438:
4418:
4415:
4395:
4392:
4369:
4366:
4363:
4362:
4350:
4325:
4321:
4318:
4314:
4307:
4303:
4300:
4296:
4289:
4282:
4278:
4275:
4271:
4264:
4260:
4257:
4253:
4240:and therefore
4225:
4221:
4218:
4214:
4207:
4203:
4200:
4196:
4189:
4182:
4178:
4175:
4171:
4164:
4160:
4157:
4153:
4128:
4124:
4121:
4117:
4113:
4109:
4105:
4102:
4098:
4077:
4074:
4054:
4051:
4028:
4025:
4022:
4021:
4009:
3985:
3981:
3978:
3974:
3967:
3963:
3960:
3956:
3949:
3942:
3938:
3935:
3931:
3924:
3920:
3917:
3913:
3882:
3878:
3875:
3871:
3863:
3859:
3856:
3852:
3845:
3841:
3838:
3834:
3829:
3825:
3822:
3818:
3810:
3803:
3799:
3796:
3792:
3784:
3780:
3777:
3773:
3766:
3762:
3759:
3755:
3750:
3746:
3743:
3739:
3713:
3709:
3706:
3702:
3680:
3676:
3673:
3669:
3642:
3638:
3635:
3631:
3624:
3620:
3617:
3613:
3606:
3599:
3595:
3592:
3588:
3581:
3577:
3574:
3570:
3541:
3537:
3534:
3530:
3523:
3519:
3516:
3512:
3505:
3498:
3494:
3491:
3487:
3480:
3476:
3473:
3469:
3435:
3431:
3428:
3424:
3419:
3415:
3412:
3408:
3401:
3397:
3394:
3390:
3385:
3381:
3378:
3374:
3367:
3360:
3356:
3353:
3349:
3344:
3340:
3337:
3333:
3326:
3322:
3319:
3315:
3310:
3306:
3303:
3299:
3271:
3267:
3264:
3260:
3255:
3251:
3248:
3244:
3237:
3233:
3230:
3226:
3221:
3217:
3214:
3210:
3203:
3196:
3192:
3189:
3185:
3180:
3176:
3173:
3169:
3162:
3158:
3155:
3151:
3146:
3142:
3139:
3135:
3106:
3097:
3062:
3058:
3055:
3052:
3049:
3045:
3038:
3034:
3031:
3028:
3025:
3021:
3014:
3007:
3003:
3000:
2997:
2994:
2990:
2983:
2979:
2976:
2973:
2970:
2966:
2938:
2934:
2931:
2928:
2925:
2921:
2914:
2910:
2907:
2904:
2901:
2897:
2890:
2883:
2879:
2876:
2873:
2870:
2866:
2859:
2855:
2852:
2849:
2846:
2842:
2815:
2811:
2808:
2805:
2802:
2798:
2794:
2790:
2786:
2783:
2780:
2777:
2773:
2763:and therefore
2751:
2747:
2744:
2741:
2738:
2734:
2730:
2726:
2722:
2719:
2716:
2713:
2709:
2688:
2685:
2682:
2679:
2659:
2656:
2653:
2650:
2630:
2627:
2624:
2621:
2618:
2591:
2587:
2583:
2556:
2553:
2548:
2545:
2536:
2533:
2530:
2529:
2517:
2494:
2491:
2469:
2465:
2462:
2458:
2451:
2447:
2444:
2440:
2435:
2431:
2428:
2424:
2417:
2413:
2409:
2406:
2402:
2380:
2376:
2373:
2369:
2347:
2343:
2340:
2336:
2314:
2310:
2307:
2303:
2289:
2286:
2285:
2284:
2265:
2262:
2248:
2235:
2232:
2221:
2215:
2210:
2206:
2203:
2200:
2194:
2191:
2177:
2176:
2157:
2154:
2149:
2137:
2131:
2120:
2117:
2114:
2100:
2099:
2096:
2093:
2090:
2054:
2051:
2049:
2046:
2028:
2025:
2022:
2000:
1996:
1993:
1968:
1964:
1961:
1934:
1931:
1928:
1906:
1902:
1899:
1876:
1873:
1870:
1848:
1844:
1841:
1826:
1823:
1820:
1819:
1807:
1778:
1775:
1771:
1750:
1726:
1725:
1720:
1715:
1701:
1698:
1696:
1693:
1673:
1669:
1665:
1661:
1655:
1649:
1646:
1640:
1634:
1631:
1625:
1620:
1617:
1611:
1608:
1602:
1596:
1593:
1587:
1584:
1581:
1578:
1572:
1566:
1560:
1557:
1551:
1546:
1540:
1537:
1531:
1528:
1525:
1519:
1513:
1507:
1504:
1498:
1493:
1487:
1484:
1478:
1475:
1472:
1449:
1443:
1440:
1434:
1428:
1424:
1420:
1416:
1412:
1409:
1403:
1400:
1394:
1391:
1368:
1365:
1359:
1356:
1353:
1347:
1344:
1338:
1335:
1332:
1329:
1323:
1320:
1314:
1308:
1305:
1299:
1296:
1293:
1269:
1266:
1241:
1238:
1236:
1233:
1228:
1227:
1212:
1208:
1205:
1201:
1194:
1190:
1187:
1183:
1176:
1169:
1165:
1162:
1158:
1151:
1147:
1144:
1140:
1112:
1108:
1105:
1101:
1094:
1090:
1087:
1083:
1076:
1069:
1065:
1062:
1058:
1051:
1047:
1044:
1040:
1013:
993:
973:
953:
933:
913:
893:
873:
850:
830:
810:
790:
770:
750:
730:
702:
699:
698:
697:
681:
677:
674:
670:
663:
659:
656:
652:
645:
638:
634:
631:
627:
620:
616:
613:
609:
594:
579:
575:
572:
568:
561:
557:
554:
550:
543:
536:
532:
529:
525:
518:
514:
511:
507:
500:
493:
489:
486:
482:
475:
471:
468:
464:
448:
433:
429:
426:
422:
415:
411:
408:
404:
397:
390:
386:
383:
379:
372:
368:
365:
361:
333:
329:
326:
322:
315:
311:
308:
304:
297:
290:
286:
283:
279:
272:
268:
265:
261:
233:
229:
226:
222:
215:
211:
208:
204:
197:
190:
186:
183:
179:
172:
168:
165:
161:
96:
93:
15:
9:
6:
4:
3:
2:
6935:
6924:
6921:
6919:
6916:
6915:
6913:
6900:
6899:
6894:
6887:
6886:
6873:
6863:
6860:
6858:
6855:
6853:
6850:
6848:
6845:
6843:
6840:
6838:
6835:
6833:
6830:
6829:
6827:
6823:
6815:
6812:
6811:
6810:
6807:
6803:
6800:
6799:
6798:
6795:
6791:
6788:
6787:
6786:
6783:
6779:
6776:
6775:
6774:
6771:
6767:
6764:
6763:
6762:
6759:
6755:
6752:
6751:
6750:
6747:
6743:
6740:
6739:
6738:
6735:
6731:
6728:
6727:
6726:
6723:
6719:
6715:
6714:
6713:
6712:
6708:
6707:
6705:
6701:
6695:
6692:
6690:
6687:
6685:
6682:
6680:
6677:
6675:
6672:
6670:
6667:
6665:
6662:
6661:
6658:
6655:
6651:
6645:
6642:
6640:
6637:
6635:
6632:
6631:
6629:
6625:
6615:
6612:
6610:
6607:
6605:
6602:
6600:
6597:
6595:
6594:
6589:
6587:
6584:
6582:
6579:
6577:
6574:
6572:
6569:
6567:
6564:
6562:
6559:
6557:
6554:
6553:
6551:
6547:
6541:
6538:
6537:
6535:
6533:
6529:
6523:
6520:
6518:
6515:
6513:
6510:
6508:
6505:
6503:
6502:Pons asinorum
6500:
6498:
6495:
6493:
6490:
6488:
6485:
6483:
6480:
6478:
6475:
6473:
6472:Hinge theorem
6470:
6468:
6465:
6463:
6460:
6458:
6455:
6453:
6450:
6448:
6445:
6443:
6440:
6439:
6437:
6435:
6434:
6428:
6425:
6421:
6415:
6412:
6410:
6407:
6405:
6402:
6400:
6397:
6395:
6392:
6390:
6387:
6385:
6382:
6380:
6377:
6375:
6372:
6370:
6367:
6365:
6362:
6360:
6357:
6355:
6352:
6350:
6347:
6345:
6342:
6340:
6337:
6333:
6330:
6328:
6325:
6324:
6323:
6320:
6318:
6315:
6311:
6308:
6307:
6306:
6303:
6299:
6296:
6294:
6291:
6290:
6289:
6286:
6285:
6283:
6277:
6271:
6268:
6264:
6261:
6259:
6256:
6254:
6251:
6250:
6249:
6246:
6245:
6243:
6239:
6233:
6232:
6228:
6226:
6225:
6221:
6219:
6215:
6211:
6209:
6205:
6201:
6199:
6198:
6194:
6192:
6191:
6187:
6185:
6184:
6180:
6178:
6177:
6173:
6171:
6167:
6163:
6161:
6157:
6153:
6151:
6147:
6143:
6141:
6139:(Aristarchus)
6137:
6133:
6131:
6130:
6126:
6124:
6123:
6119:
6117:
6113:
6109:
6107:
6103:
6099:
6097:
6096:
6092:
6090:
6086:
6082:
6080:
6079:
6075:
6073:
6070:
6068:
6067:
6063:
6062:
6060:
6056:
6050:
6047:
6045:
6044:Zeno of Sidon
6042:
6040:
6037:
6035:
6032:
6030:
6027:
6025:
6022:
6020:
6017:
6015:
6012:
6010:
6007:
6005:
6002:
6000:
5997:
5995:
5992:
5990:
5987:
5985:
5982:
5980:
5977:
5975:
5972:
5970:
5967:
5965:
5962:
5960:
5957:
5955:
5952:
5950:
5947:
5945:
5942:
5940:
5937:
5935:
5932:
5930:
5927:
5925:
5922:
5920:
5917:
5915:
5912:
5910:
5907:
5905:
5902:
5900:
5897:
5895:
5892:
5890:
5887:
5885:
5882:
5880:
5877:
5875:
5872:
5870:
5867:
5865:
5862:
5860:
5857:
5855:
5852:
5850:
5847:
5845:
5842:
5840:
5837:
5835:
5832:
5830:
5827:
5825:
5822:
5820:
5817:
5815:
5812:
5810:
5807:
5805:
5802:
5800:
5797:
5795:
5792:
5790:
5787:
5785:
5782:
5780:
5777:
5775:
5772:
5770:
5767:
5765:
5762:
5760:
5757:
5755:
5752:
5750:
5747:
5745:
5742:
5740:
5737:
5735:
5732:
5730:
5727:
5725:
5722:
5720:
5717:
5715:
5712:
5710:
5707:
5705:
5702:
5700:
5697:
5695:
5692:
5690:
5687:
5685:
5682:
5680:
5677:
5676:
5674:
5672:
5668:
5664:
5660:
5653:
5648:
5646:
5641:
5639:
5634:
5633:
5630:
5624:
5621:
5619:
5615:
5614:
5609:
5608:
5603:
5601:
5597:
5596:
5592:
5591:
5582:
5581:9783662530344
5578:
5574:
5570:
5567:
5563:
5562:
5555:
5549:
5545:
5540:
5539:
5532:
5529:
5525:
5524:
5517:
5511:
5507:
5503:
5502:
5496:
5493:
5489:
5488:
5481:
5479:0-8218-4347-8
5475:
5471:
5467:
5463:
5460:
5456:
5455:
5448:
5446:9780826473622
5442:
5438:
5434:
5430:
5429:
5419:
5415:
5409:
5402:
5401:9780521118019
5398:
5394:
5388:
5381:
5375:
5368:
5364:
5360:
5354:
5348:, pp. 214–217
5347:
5346:9783030409746
5343:
5339:
5333:
5326:
5324:
5319:
5317:
5312:
5308:
5307:
5302:
5298:
5294:
5290:
5286:
5281:
5277:
5271:
5264:
5260:
5259:
5252:
5246:
5242:
5237:
5236:
5227:
5219:
5217:3-528-07243-1
5213:
5209:
5202:
5195:
5193:0-486-42515-0
5189:
5185:
5181:
5175:
5156:
5152:
5148:
5144:
5138:
5131:
5124:
5123:9783662530344
5120:
5116:
5112:
5106:
5104:
5102:
5094:
5090:
5089:
5082:
5080:0-8218-4347-8
5076:
5072:
5068:
5062:
5055:
5051:
5050:Strahlensätze
5046:
5044:
5035:
5033:3-506-99189-2
5029:
5025:
5018:
5016:
5011:
5000:
4987:
4967:
4964:
4944:
4941:
4919:
4915:
4911:
4908:
4888:
4868:
4846:
4842:
4821:
4791:
4788:
4773:
4770:
4757:
4754:
4743:
4733:
4729:
4725:
4690:
4687:
4672:
4669:
4656:
4653:
4642:
4634:
4631:
4600:
4597:
4589:
4581:
4578:
4570:
4562:
4559:
4551:
4543:
4540:
4515:
4512:
4507:
4503:
4482:
4479:
4459:
4439:
4436:
4416:
4413:
4393:
4390:
4378:
4373:
4361:
4348:
4340:
4319:
4316:
4301:
4298:
4287:
4276:
4273:
4258:
4255:
4219:
4216:
4201:
4198:
4187:
4176:
4173:
4158:
4155:
4122:
4119:
4111:
4103:
4100:
4075:
4072:
4052:
4049:
4037:
4032:
4020:
4007:
3979:
3976:
3961:
3958:
3947:
3936:
3933:
3918:
3915:
3897:
3876:
3873:
3857:
3854:
3839:
3836:
3823:
3820:
3808:
3797:
3794:
3778:
3775:
3760:
3757:
3744:
3741:
3707:
3704:
3674:
3671:
3657:
3636:
3633:
3618:
3615:
3604:
3593:
3590:
3575:
3572:
3535:
3532:
3517:
3514:
3503:
3492:
3489:
3474:
3471:
3453:
3450:
3429:
3426:
3413:
3410:
3395:
3392:
3379:
3376:
3365:
3354:
3351:
3338:
3335:
3320:
3317:
3304:
3301:
3265:
3262:
3249:
3246:
3231:
3228:
3215:
3212:
3201:
3190:
3187:
3174:
3171:
3156:
3153:
3140:
3137:
3121:
3104:
3095:
3077:
3056:
3053:
3050:
3032:
3029:
3026:
3012:
3001:
2998:
2995:
2977:
2974:
2971:
2932:
2929:
2926:
2908:
2905:
2902:
2888:
2877:
2874:
2871:
2853:
2850:
2847:
2828:
2809:
2806:
2803:
2792:
2784:
2781:
2778:
2745:
2742:
2739:
2728:
2720:
2717:
2714:
2686:
2683:
2680:
2657:
2654:
2651:
2628:
2625:
2622:
2619:
2616:
2608:
2604:
2585:
2572:
2565:
2560:
2552:
2544:
2542:
2525:
2521:
2513:
2509:
2507:
2501:
2498:
2490:
2488:
2463:
2460:
2445:
2442:
2429:
2426:
2415:
2407:
2404:
2374:
2371:
2341:
2338:
2308:
2305:
2263:
2260:
2246:
2233:
2230:
2219:
2213:
2208:
2204:
2201:
2198:
2192:
2189:
2182:
2181:
2180:
2155:
2152:
2147:
2135:
2129:
2118:
2115:
2112:
2105:
2104:
2103:
2097:
2094:
2091:
2088:
2087:
2086:
2082:
2080:
2076:
2067:
2059:
2045:
2043:
2026:
2023:
2020:
1994:
1991:
1962:
1959:
1948:
1932:
1929:
1926:
1900:
1897:
1874:
1871:
1868:
1842:
1839:
1818:
1816:
1812:
1806:
1804:
1800:
1795:
1792:
1776:
1773:
1769:
1748:
1740:
1736:
1732:
1724:
1721:
1719:
1716:
1714:
1711:
1710:
1709:
1707:
1692:
1690:
1685:
1667:
1659:
1644:
1638:
1629:
1606:
1600:
1591:
1582:
1579:
1570:
1555:
1535:
1529:
1526:
1517:
1502:
1482:
1476:
1473:
1438:
1426:
1418:
1410:
1398:
1392:
1363:
1357:
1354:
1351:
1342:
1336:
1333:
1330:
1318:
1312:
1303:
1294:
1291:
1283:
1279:
1275:
1265:
1263:
1259:
1255:
1246:
1232:
1206:
1203:
1188:
1185:
1174:
1163:
1160:
1145:
1142:
1106:
1103:
1088:
1085:
1074:
1063:
1060:
1045:
1042:
1027:
1026:
1025:
1011:
991:
971:
964:, such that
951:
931:
911:
891:
871:
862:
848:
828:
808:
788:
768:
748:
728:
715:
707:
675:
672:
657:
654:
643:
632:
629:
614:
611:
595:
573:
570:
555:
552:
541:
530:
527:
512:
509:
498:
487:
484:
469:
466:
449:
427:
424:
409:
406:
395:
384:
381:
366:
363:
327:
324:
309:
306:
295:
284:
281:
266:
263:
227:
224:
209:
206:
195:
184:
181:
166:
163:
146:
145:
144:
127:
123:
119:
115:
109:
101:
92:
90:
89:
84:
80:
76:
72:
69:
65:
61:
57:
53:
52:line segments
49:
45:
41:
37:
33:
29:
22:
6889:
6876:
6718:Thomas Heath
6709:
6592:
6576:Law of sines
6481:
6432:
6364:Golden ratio
6229:
6222:
6213:
6207:(Theodosius)
6203:
6195:
6188:
6181:
6174:
6165:
6155:
6149:(Hipparchus)
6145:
6135:
6127:
6120:
6111:
6101:
6093:
6088:(Apollonius)
6084:
6076:
6064:
6039:Zeno of Elea
5799:Eratosthenes
5789:Dionysodorus
5618:Cut-the-Knot
5612:
5606:
5594:
5572:
5566:Google Books
5559:
5537:
5528:Google Books
5526:, p. 34, at
5521:
5500:
5492:Google Books
5490:, p. 10, at
5485:
5469:
5459:Google Books
5457:, p. 84, at
5452:
5436:
5433:French, Doug
5413:
5408:
5392:
5387:
5379:
5374:
5358:
5353:
5337:
5332:
5322:
5315:
5305:
5303:". (Source:
5300:
5295:". However,
5292:
5288:
5270:
5263:Google Books
5256:
5234:
5226:
5207:
5201:
5183:
5174:
5154:
5150:
5146:
5142:
5130:
5114:
5093:Google Books
5091:, p. 10, at
5086:
5070:
5061:
5053:
5049:
5023:
4382:
4341:
4041:
3898:
3658:
3454:
3451:
3122:
3078:
2829:
2606:
2605:
2570:
2569:
2550:
2538:
2519:
2503:
2496:
2291:
2178:
2101:
2083:
2072:
1946:
1828:
1810:
1809:
1798:
1797:
1727:
1703:
1695:Applications
1686:
1274:vector space
1272:In a normed
1271:
1251:
1229:
863:
720:
141:
125:
121:
117:
113:
86:
39:
35:
31:
27:
25:
6785:mathematics
6593:Arithmetica
6190:Ostomachion
6159:(Autolycus)
6078:Arithmetica
5854:Hippocrates
5784:Dinostratus
5769:Dicaearchus
5699:Aristarchus
5564:, p. 3, at
5561:online copy
5523:online copy
5487:online copy
5454:online copy
5403:, pp. 17-18
5369:, pp. 27-36
5261:, p. 7, at
5258:online copy
5088:online copy
4472:intersects
75:Babylonians
6912:Categories
6837:Babylonian
6737:arithmetic
6703:History of
6532:Apollonius
6217:(Menelaus)
6176:On Spirals
6095:Catoptrics
6034:Xenocrates
6029:Thymaridas
6014:Theodosius
5999:Theaetetus
5979:Simplicius
5969:Pythagoras
5954:Posidonius
5939:Philonides
5899:Nicomachus
5894:Metrodorus
5884:Menaechmus
5839:Hipparchus
5829:Heliodorus
5779:Diophantus
5764:Democritus
5744:Chrysippus
5714:Archimedes
5709:Apollonius
5679:Anaxagoras
5671:(timeline)
5600:PlanetMath
5426:References
5285:Hieronymus
1262:conversely
1254:similarity
6298:Inscribed
6058:Treatises
6049:Zenodorus
6009:Theodorus
5984:Sosigenes
5929:Philolaus
5914:Oenopides
5909:Nicoteles
5904:Nicomedes
5864:Hypsicles
5759:Ctesibius
5749:Cleomedes
5734:Callippus
5719:Autolycus
5704:Aristotle
5684:Anthemius
5182:(2003) ,
4988:◻
4513:≠
4349:◻
4008:◻
3096:⋅
3048:△
3024:△
2993:△
2969:△
2924:△
2900:△
2869:△
2845:△
2801:△
2776:△
2737:△
2712:△
2678:△
2649:△
2623:∥
2586:…
2571:Notation:
2231:⋅
2202:⋅
1999:¯
1967:¯
1905:¯
1847:¯
1774:−
1668:λ
1654:‖
1648:→
1633:→
1624:‖
1619:‖
1610:→
1595:→
1583:⋅
1580:λ
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1565:‖
1559:→
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1527:λ
1524:‖
1512:‖
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1474:λ
1471:‖
1448:‖
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1408:‖
1402:→
1393:λ
1390:‖
1367:→
1358:⋅
1355:λ
1346:→
1337:⋅
1334:λ
1322:→
1307:→
1295:⋅
1292:λ
79:Egyptians
60:parallels
6862:Japanese
6847:Egyptian
6790:timeline
6778:timeline
6766:timeline
6761:geometry
6754:timeline
6749:calculus
6742:timeline
6730:timeline
6433:Elements
6279:Concepts
6241:Problems
6214:Spherics
6204:Spherics
6169:(Euclid)
6115:(Euclid)
6112:Elements
6105:(Euclid)
6066:Almagest
5974:Serenus
5949:Porphyry
5889:Menelaus
5844:Hippasus
5819:Eutocius
5794:Domninus
5689:Archytas
5435:(2004).
5311:MacTutor
5297:Plutarch
4528:. Since
4452:through
3725:in (a):
3556:and (b)
3100:altitude
3092:baseline
88:Elements
6842:Chinese
6797:numbers
6725:algebra
6653:Related
6627:Centers
6423:Results
6293:Central
5964:Ptolemy
5959:Proclus
5924:Perseus
5879:Marinus
5859:Hypatia
5849:Hippias
5824:Geminus
5814:Eudoxus
5804:Eudemus
5774:Diocles
5309:of the
5208:Algebra
4383:Assume
4368:Claim 3
4027:Claim 2
2555:Claim 1
2039:ratio.
44:theorem
6857:Indian
6634:Cyrene
6166:Optics
6085:Conics
6004:Theano
5994:Thales
5989:Sporus
5934:Philon
5919:Pappus
5809:Euclid
5739:Carpus
5729:Bryson
5579:
5550:
5512:
5476:
5443:
5399:
5365:
5344:
5318:, 147A
5247:
5214:
5190:
5157:|
5141:|
5121:
5077:
5052:. In:
5030:
2609:Since
2607:Proof:
2267:
2250:
2237:
2223:
2159:
2139:
2122:
2075:Thales
1430:
1278:axioms
1276:, the
128:|
112:|
83:Euclid
71:Thales
6852:Incan
6773:logic
6549:Other
6317:Chord
6310:Axiom
6288:Angle
5944:Plato
5834:Heron
5754:Conon
5418:JSTOR
5280:Pliny
5007:Notes
2547:Proof
2264:146.7
2013:in a
1861:in a
1004:than
6814:list
6102:Data
5874:Leon
5724:Bion
5577:ISBN
5548:ISBN
5546:–7.
5510:ISBN
5474:ISBN
5441:ISBN
5397:ISBN
5363:ISBN
5342:ISBN
5320:and
5278:and
5245:ISBN
5212:ISBN
5188:ISBN
5119:ISBN
5075:ISBN
5028:ISBN
4957:and
4834:and
4406:and
3692:and
3455:(a)
3286:and
2953:and
2670:and
2220:1.63
1382:and
944:and
821:and
77:and
56:rays
26:The
6716:by
6430:In
5616:at
5598:at
4814:So
4495:in
2508:).
2234:180
2156:180
2136:230
841:at
85:'s
46:in
38:or
6914::
5508:.
5506:34
5243:.
5165:BD
5161:AC
5155:SD
5151:SC
5147:SB
5143:SA
5100:^
5042:^
5014:^
4812:.
2484:.
2119:65
1708::
1127:,
348:,
248:,
136:BD
132:AC
126:BD
122:AC
118:SB
114:SA
91:.
34:,
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5637:v
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4968:D
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4889:S
4869:S
4847:0
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4762:|
4758:D
4755:S
4751:|
4744:=
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4639:|
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4594:|
4590::
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4567:|
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4560:S
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4508:0
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4320:D
4317:B
4313:|
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4299:A
4295:|
4288:=
4281:|
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4274:S
4270:|
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4112:=
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3948:=
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3809:=
3802:|
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3605:=
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3511:|
3504:=
3497:|
3493:D
3490:C
3486:|
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3468:|
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3427:E
3423:|
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3411:S
3407:|
3400:|
3396:C
3393:E
3389:|
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3380:A
3377:S
3373:|
3366:=
3359:|
3355:F
3352:A
3348:|
3343:|
3339:D
3336:S
3332:|
3325:|
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3318:A
3314:|
3309:|
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3298:|
3270:|
3266:C
3263:E
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3247:A
3243:|
3236:|
3232:C
3229:E
3225:|
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3213:S
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3202:=
3195:|
3191:F
3188:A
3184:|
3179:|
3175:D
3172:C
3168:|
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3154:A
3150:|
3145:|
3141:C
3138:S
3134:|
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3061:|
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3030:C
3027:S
3020:|
3013:=
3006:|
3002:A
2999:D
2996:S
2989:|
2982:|
2978:A
2975:C
2972:S
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2930:B
2927:C
2920:|
2913:|
2909:A
2906:C
2903:S
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2889:=
2882:|
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2875:D
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2810:A
2807:D
2804:S
2797:|
2793:=
2789:|
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2782:C
2779:S
2772:|
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2740:C
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2725:|
2721:A
2718:D
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2708:|
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2684:B
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2582:|
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2401:|
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2372:F
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2335:|
2326:,
2313:|
2309:F
2306:C
2302:|
2271:m
2261:=
2254:m
2247:2
2241:m
2227:m
2214:=
2209:B
2205:A
2199:C
2193:=
2190:D
2163:m
2153:=
2148:2
2143:m
2130:+
2126:m
2116:=
2113:C
2027:3
2024::
2021:5
1995:B
1992:A
1963:B
1960:A
1947:m
1933:n
1930:+
1927:m
1901:B
1898:A
1875:n
1872::
1869:m
1843:B
1840:A
1777:1
1770:a
1749:a
1672:|
1664:|
1660:=
1645:b
1639:+
1630:a
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1607:b
1601:+
1592:a
1586:(
1571:=
1556:b
1536:b
1518:=
1503:a
1483:a
1439:a
1423:|
1415:|
1411:=
1399:a
1364:b
1352:+
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1331:=
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1319:b
1313:+
1304:a
1298:(
1211:|
1207:D
1204:F
1200:|
1193:|
1189:F
1186:B
1182:|
1175:=
1168:|
1164:C
1161:E
1157:|
1150:|
1146:E
1143:A
1139:|
1111:|
1107:D
1104:F
1100:|
1093:|
1089:C
1086:E
1082:|
1075:=
1068:|
1064:F
1061:B
1057:|
1050:|
1046:E
1043:A
1039:|
1012:E
992:S
972:F
952:F
932:E
912:S
892:S
872:S
849:S
829:C
809:A
789:S
769:S
749:S
729:S
680:|
676:D
673:C
669:|
662:|
658:C
655:S
651:|
644:=
637:|
633:B
630:A
626:|
619:|
615:A
612:S
608:|
578:|
574:D
571:B
567:|
560:|
556:C
553:A
549:|
542:=
535:|
531:D
528:S
524:|
517:|
513:C
510:S
506:|
499:=
492:|
488:B
485:S
481:|
474:|
470:A
467:S
463:|
432:|
428:D
425:S
421:|
414:|
410:C
407:S
403:|
396:=
389:|
385:B
382:S
378:|
371:|
367:A
364:S
360:|
332:|
328:D
325:C
321:|
314:|
310:D
307:S
303:|
296:=
289:|
285:B
282:A
278:|
271:|
267:B
264:S
260:|
232:|
228:D
225:C
221:|
214:|
210:C
207:S
203:|
196:=
189:|
185:B
182:A
178:|
171:|
167:A
164:S
160:|
138:.
23:.
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