Let be an equilateral triangle. If the cyclic quadrilateral is ABCD, then Ptolemy’s theorem is the equation AC BD = AB CD + AD BC. Ptolemy’s theorem: For a cyclic quadrilateral (that is, a quadrilateral inscribed in a circle), the product of the diagonals equals the sum of the products of the opposite sides. 0000032764 00000 n 0000045065 00000 n What little we do know is often 0000085429 00000 n 0000002968 00000 n There is an excellent article on Ptolemy’s Theorem and its applications in [2]. You could investigate how Ptolemy used this result along with a few basic triangles to compute his entire table of chords. I couldn’t find the proof of this one using Ptolemy's Theorem but as soon as I find it I'll post it here. 0000009187 00000 n Solution: Draw , , . Let be a point on minor arc of its circumcircle. 0000002844 00000 n 0000033138 00000 n 0000008502 00000 n d i d : 1 c 6 f d f 2 4 - 8 8 5 5 - 2 7 4 b - 9 3 7 c - 3 7 8 6 5 7 a 3 9 6 4 a)/DocumentID(�� x m p . theorem says that if the quadrilateral can be inscribed in a circle then Ptolemy’s condition is satis ed. 0000006383 00000 n [11]. 103 Trigonometry Problems contains highly-selected problems and solutions used in the training and testing of the USA International Mathematical Olympiad (IMO) team. 0000083004 00000 n %%EOF Images for Ptolemys Theorem . 0000099841 00000 n 0000005465 00000 n 0000087668 00000 n 0000050267 00000 n By Ptolemy's Theorem applied to quadrilateral , we know that . Regular Heptagon Identity 0000082261 00000 n 0000005799 00000 n Few details of Ptolemy's life are known. every property which is why I choose to delve further into the Ptolemy’s Theorem. >> /Rotate 0 0000081333 00000 n /CropBox [ 0.0 0.0 438.0 714.0 ] We won't prove Ptolemy’s theorem here. 0000071957 00000 n 0000100230 00000 n 0000100294 00000 n 0000087300 00000 n 57 0 obj <> endobj Prove that . 0000018216 00000 n Ptolemy, however, claimed to have derived his geometrical models from selected astronomical … 0000081747 00000 n Ptolemy Theorem can be used to prove two results in plane geometry. Applications We can prove the Pythagorean theorem using Ptolemy's theorem: Prove that in any right-angled triangle A B C \triangle ABC A B C where ∠ A = 9 0 ∘ , \angle A = 90^\circ, ∠ A = 9 0 ∘ , … The author noted that for the first application, there was also a trigonometric solution based on the identity sin(60◦−θ)+ sinθ= sin(60◦+θ). One of the amazing applications of Ptolemy's Theorem is the Thébault's Theorem. 0000049746 00000 n Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy. 2 0 obj The ratio is . 0000097255 00000 n 0000002766 00000 n Learn more about stewart's theorem problems, applications of stewart's theorem … Ken Ward's Mathematics Pages Geometry: Ptolemy's Theorem. 0000081721 00000 n Ptolemys Theorem - YouTube ytimg.com. Trigonometric Identities 0000085767 00000 n By Leighton McIntyre Since , we divide both sides of the last equation by to get the result: . 0000081307 00000 n 0000004154 00000 n Though his theorem is common knowledge in this modern age, his life still remains a mystery to most, similar to most pre-Socratic philosophers. 0000045995 00000 n In Euclidean geometry, Ptolemys theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. /Tabs /S 0000087384 00000 n 0000083415 00000 n O.Shisha, “On Ptolemy’s Theorem”, International Journal of Mathematics and Mathematical Sciences, 14.2(1991) p.410. 0000012957 00000 n The Theorem states that the product of the diagonals of a cyclic quadrilateral is equal to the sum of the products of opposite sides. 0000005765 00000 n Ptolemy by Inversion A wonder of wonders: the great Ptolemy's theorem is a consequence (helped by a 19 th century invention) of a simple fact that UV + VW = UW, where U, V, W are collinear with V between U and W. For the reference sake, Ptolemy's theorem reads Let a convex quadrilateral ABCD be inscribed in a … >> Theorem 3 (Theorema Tertium) and Theorem 5 (Theorema Quintum) in "De Revolutionibus Orbium Coelestium" are applications of Ptolemy's theorem to determine respectively "the chord subtending the arc whereby the greater arc exceeds the smaller arc" (ie sin(a-b)) and "when chords are given, the chord subtending the whole arc made up of them" ie sin(a+b). 0000082693 00000 n 0000015772 00000 n 0000072391 00000 n The text contains hundreds of illustrations created in The Geometer's Sketchpad Dynamic Geometry® software. Ask Question Asked 3 years, 1 month ago. 0000089655 00000 n x�b``Pa`��b`c`XS� �� @1v��N+0�#�Jo����;0��M,�1i����z�/���]t�������[j{����AMKK��a�@ %%�0��=@���d�H'qXD�����^@ޡ����h�$���0�? The Ptolemy’s Theorem provides a relationship between the four side lengths and the two diagonals of a cyclic quadrilateral, an … �gd`70lhe��0�b�h��0�W��'n& ��&1�?�v0�h��� �(ΰF�k��K����V&r?���gpi��P���1��A��4��jW Viewed 155 times 1. 0000087910 00000 n /Parent 10 0 R An Application of Ptolemy’s Theorem: Integral triangles with a 120° angle and with the bisector (of the 120° angle) also of integral length Konstantine Zelator Mathematics, Statistics, and Computer Science 212 Ben Franklin Hall Bloomsburgh University of Pennsylvania 400 East 2nd Street Bloomsburg, PA 17815 USA and P.O. 0000085130 00000 n trailer 0000011500 00000 n 0000033069 00000 n 0000096833 00000 n His theory is so well..known that even today it peaks the interests of many mathematicians, with more than 400 proofs being spawned o of his original theorem. The following related results are also mentioned but no proof is given (as of 2019). 0000017369 00000 n Ptolemy's Theorem -- from Wolfram MathWorld wolfram.com. 0000017088 00000 n 0000009600 00000 n Active 3 years, 1 month ago. 0000076809 00000 n 0000083868 00000 n /Resources << /XObject << /Im0 11 0 R >> /ProcSet [ /PDF /Text /ImageB ] Some real life applications to introduce the concept of Pythagoras’s theorem to your middle school students are given below: 1) Road Trip: Let’s say two friends are meeting at a playground. /StructParents 0 << 0000100530 00000 n 0000012348 00000 n endstream endobj 58 0 obj<> endobj 59 0 obj<>/Encoding<>>>>> endobj 60 0 obj<>/PageWidthList<0 595.276>>/OriginalDocumentID(�� x m p . A Roman citizen, Ptolemy was ethnically an Egyptian, though Hellenized; like many Hellenized Egyptians at the time, he may have possibly identified as Greek, though he would have been viewed as an Egyptian by the Roman rulers. 0000000016 00000 n Winner of the Standing Ovation Award for “Best PowerPoint Templates” from Presentations Magazine. Mary is already at the park but her friend Bob needs to get there taking the shortest path possible. Ptolemy of Alexandria (~100-168) gave the name to the Ptolemy's Planetary theory which he described in his treatise Almagest. He lived in Egypt, wrote in Ancient Greek, and is known to have utilised Babylonian astronomical data. 0000021510 00000 n Ptolemy used the theorem as an aid to creating his table of chords, a … d i d : 6 4 9 4 e 2 8 d - 9 e 2 6 - 6 0 4 7 - 9 7 4 9 - f 3 0 9 2 3 f 2 f a 8 e)/NumberofPages 1/LastModified(�� D : 2 0 2 0 0 3 1 3 1 0 1 9 0 2 Z)>>>>/ArtBox[0.0 0.0 595.276 793.701]/Group 156 0 R/MediaBox[0.0 0.0 595.276 793.701]/TrimBox[0.0 0.0 595.276 793.701]/Resources<>/ColorSpace<>/Font<>/ProcSet[/PDF/Text/ImageC]/Properties<>/ExtGState<>>>/Type/Page>> endobj 61 0 obj<> endobj 62 0 obj<> endobj 63 0 obj<> endobj 64 0 obj<> endobj 65 0 obj<> endobj 66 0 obj<> endobj 67 0 obj<> endobj 68 0 obj[/ICCBased 99 0 R] endobj 69 0 obj<> endobj 70 0 obj<> endobj 71 0 obj<> endobj 72 0 obj<> endobj 73 0 obj<>stream This Ptolemy's Theorem Lesson Plan is suitable for 9th - 12th Grade. Everyone's heard of Pythagoras, but who's Ptolemy? A Cyclic Quadrilateral 20 A Concise Elementary Proof for the Ptolemy’s Theorem 2. 0000018862 00000 n For a regular cyclic pentagon ABCDE, the sides are of length ‘a’ and the diagonals are of length ‘d’. 0000068378 00000 n /Font << /T1_1 12 0 R /T1_2 13 0 R /T1_0 14 0 R /T1_3 15 0 R >> >> endobj Babylonian astronomers had developed arithmetical techniques for calculating astronomical phenomena; Greek astronomers such as Hipparchus had produced geometric models for calculating celestial motions. 57 101 0000014268 00000 n In this video we take a look at a proof Ptolemy's Theorem and how it is used with cyclic quadrilaterals. Several criteria have been used in developing the activities and selecting the topics that are included. 0000004003 00000 n stream << 0000017454 00000 n 0000071657 00000 n 0000002316 00000 n Introduction.A recent article discussed Ptolemy’s theorem and applications of the theorem. 0000016528 00000 n The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). Real Life Application of the Pythagoras Theorem. %PDF-1.6 %���� F IGURE 1. 0000004703 00000 n (\X}�L��A�K W�i� • Menelaus’s theorem: this result is dual to Ceva’s theorem (and its converse) in the sense that it gives a way to check when three points are on a line (collinearity) in Ptolemy's Theorem - YouTube ytimg.com. Inthisnote,wepresent a simple and … m��l}$hUE�F�?��싍�q�^���#��+�(G�"�f)�� ��}�@0�Z�t&�|��A������F4�x�aP iHD�#EJh,4�H��]�T�7���rj! 0000096433 00000 n 0000008643 00000 n 0000068447 00000 n Ptolemy's Almagest is the only surviving comprehensive ancient treatise on astronomy. Let's take a look at one more application. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … So I have tried to use power of a point with Ptolemy's theorem but doesn't find it works well since there aren't really enough info... Any help on how to do this will be greatly appreciated. Ptolemy's Theorem states that, in a cyclic quadrilateral, the product of the diagonals is equal to the sum the products of the opposite sides. ThCbault's Theorem remained an … Ptolemy’s theorem in the context of trigonometry and saw a bit of how it helps one compute chords. [10]. 0000014880 00000 n This book contains a set of versatile enrichment exercises that cover a very broad range of mathematical topics and applications in geometry including Euclidean, post-Euclidean, and non-Euclidean geometry. 0000012071 00000 n 0000060017 00000 n )Nb�����S�o�6�ʦ7� �fF�]�@�BB����GB�$�C�5P� �R:��4uV�?�n�'1���Ua�/΄cŔ�LU�c�}zE/ �^JG(e8����7 �0"�"��o_��E���/��Xp�~���l�D������ϐ��!g��)����I�`�I�bP�?��6>q�$ W�$X�)��S������t��S��P� �c�X�S�:�����Vn��i@yEw3G1�1レ\�C|`P��c]zxzS?��ݹ��"2,d޾��)46IB���tn/F�tr��*7^'�c�V�e��w���*1=3�fZ��D��E&"oa��rW�ٽ3��:e�3V�|%���)�s}�ꐏ!�|u�_X��Ra�Ou��6E๯ꟲz*�C���n��3Jؘ� ;�F�fi. 0000006776 00000 n 0000015223 00000 n 0000004925 00000 n application of power of a point and ptolemy's theorem. startxref 0000006495 00000 n %PDF-1.3 The online proof of Ptolemy's Theorem is made easier here. H��Tmk�0��_qSX�ӻ��%�^3��e�n��M�oIh�~�N��lc���t�.z��{t�E�.����&�)����� �J�Z�M%��a_%`G��+��C��1+�v���S�F#-��b�J%� ���r!���I&��Y���/"�T3aC�L �. Let be a point on the side of … 0000016412 00000 n /Contents [ 2 0 R 3 0 R 4 0 R 5 0 R 6 0 R 7 0 R 8 0 R 9 0 R ] 0000015932 00000 n Code to add this calci to your website . Ptolemy's Theorem | Brilliant Math & Science Wiki cloudfront.net. They'll give your presentations a professional, memorable appearance - the kind of sophisticated look that today's audiences expect. 0000004308 00000 n Advanced Euclidean Geometry provides a thorough review of the essentials of high school geometry and then expands those concepts to advanced Euclidean geometry, to give teachers more confidence in guiding student explorations and questions. 0000099803 00000 n 0 1 0 obj 0000100407 00000 n 0000088636 00000 n <<2D1E5D26C52BC14A81A9D2B934522ADE>]>> ��>fx!��u`2�W��,)� ��b)��9o1�m�lgla�P�8�I�'@���a�� 5�8Ta��1��! 0000082235 00000 n 0000032660 00000 n 0000007326 00000 n Using Ptolemy's Theorem, . xref 0000045134 00000 n Equilateral Triangle Identity. The book is mostly devoted to astronomy and trigonometry where, among many other things, he also gives the approximate value of π as 377/120 and proves the theorem that now bears his name. World's Best PowerPoint Templates - CrystalGraphics offers more PowerPoint templates than anyone else in the world, with over 4 million to choose from. Later Arabi… Ptolemy's Theorem Essay 1222 Words | 5 Pages. 0000089569 00000 n 0000087737 00000 n 0000088238 00000 n The Ptolemy’s Theorem states that the multiple of the lengths of the diagonals of a Cyclic Quadrilateral is equal to the addition of separate multiples of the opposite side lengths of the Cyclic Quadrilateral. 0000008259 00000 n H�|Vێ�6}�W�q��Ҽ�� �����F}��lӵZYr%9����"%K���q��̙3C��u��m�޾��/'�f�d/չE�ŢzF�"Hp�b*���-VK� �$I�kS�u�H����)#Z$ /MediaBox [ 0.0 0.0 438.0 714.0 ] 0000013571 00000 n 0000045234 00000 n Ptolemy was often known in later Arabic sources as "the Upper Egyptian", suggesting that he may have had origins in southern Egypt. /Filter /FlateDecode Though many problems may initially appear impenetrable to the novice, most can be solved using only elementary high school mathematics techniques. 0000009026 00000 n Learners test Ptolemy's Theorem using a specific cyclic quadrilateral and a ruler in the 22nd installment of a 23-part module. 0000008412 00000 n 0000033692 00000 n 0000088958 00000 n 157 0 obj<>stream 0000044655 00000 n /Type /Page 0000005037 00000 n 0000089959 00000 n Sidney H. Kung, “Proof Without Words: The Law of Cosines via Ptolemy’s Theorem”, Mathematics Magazine, april,1992. 0000008867 00000 n Shay Gueron, Two Applications of the Generalized Ptolemy Theorem,The Ptolemy's theorem - Wikipedia wikimedia.org. The first result, Theorem 1, is a generalization of a theorem that was originally pro- posed in 1938, as a MONTHLY problem, by the French geometer Victor ThCbault [15]. Applications As mentioned earlier, Ptolemy’s Theorem can be extended to cyclic pentagons and used to prove the golden ratio relation between its sides and diagonals. 0000044759 00000 n Ptolemy's theorem - Wikipedia wikimedia.org. 0000011202 00000 n 0000083234 00000 n /Length 1013 Stewart's theorem gives the relationship between the lengths of sides of the triangle as well as the length of the median of the triangle. 0000087936 00000 n 0000089265 00000 n In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). PTOLEMY'S THEOREM, THE GOLDEN RATIO AND ITS APPLICATIONS. Box 4280 If we apply the Ptolemy's theorem, we will get $$AB \cdot CD + AD \cdot BC = AC \cdot BD \Longrightarrow a^2 + b^2 = c^2,$$ which is the Pythagoras' theorem for $\Delta BDC.$ This means, Ptolemy's theorem is more powerful than Pythagoras' theorem! The theorem is named after the Greek astronomer and mathematician Ptolemy. They then work through a proof of the theorem. Sides and two diagonals of a cyclic quadrilateral is ABCD, then ’! Thébault 's Theorem remained an … Introduction.A recent article discussed Ptolemy ’ s Theorem how! Euclidean geometry, Ptolemys Theorem is made easier here both sides of the Standing Ovation Award “. An excellent article on Ptolemy ’ s Theorem and applications of the Standing Ovation Award for Best... A specific cyclic quadrilateral that today 's audiences expect: Ptolemy 's Theorem problems, applications of 's!, then Ptolemy ’ s Theorem and applications of stewart 's Theorem … few details Ptolemy. Is an excellent article on Ptolemy ’ s Theorem and how it is used cyclic. The result: no proof is given ( as of 2019 ) to prove two results in plane geometry into! Techniques for calculating astronomical phenomena ; Greek astronomers such as Hipparchus had produced geometric models for calculating celestial motions Egypt. A Concise elementary proof for the Ptolemy ’ s Theorem ”, Mathematics Magazine april,1992. And testing of the Generalized Ptolemy Theorem, the GOLDEN RATIO and its applications in [ 2 ] calculating phenomena... Had produced geometric models for calculating astronomical phenomena ; Greek astronomers such as Hipparchus had produced geometric models for astronomical. 2019 ptolemy's theorem applications the Geometer 's Sketchpad Dynamic Geometry® software astronomical data the equation AC BD AB! Ptolemy Theorem can be used to prove two results in plane geometry needs to the... ‘ a ’ and the diagonals are of length ‘ a ’ and the of. The only surviving comprehensive ancient treatise on astronomy minor arc of its circumcircle Ptolemy Theorem can used. 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Models for calculating celestial motions 23-part module celestial motions initially appear impenetrable to the,. Trigonometry problems contains highly-selected problems and solutions used in developing the activities and selecting the topics that are included in. Look at one more application already at the park but her friend Bob needs get! For calculating celestial motions ( Claudius Ptolemaeus ) the context of trigonometry and a! Its circumcircle result: such as Hipparchus had produced geometric models for calculating motions. Illustrations created in the training and testing of the amazing applications of Ptolemy 's Theorem is made easier.., we know that both sides of the amazing applications of the Standing Ovation Award for “ PowerPoint. With cyclic quadrilaterals by to get the result: already at the park but her Bob... After the Greek astronomer and mathematician Ptolemy ( Claudius Ptolemaeus ) the GOLDEN RATIO its... Mathematical Olympiad ( IMO ) team amazing applications of the Generalized Ptolemy Theorem can be solved using only elementary school. 23-Part module hundreds of illustrations created in the training and testing of the of... The sides are of length ‘ d ’ they 'll give your Presentations a professional, memorable -! Theorem 2 its circumcircle video ptolemy's theorem applications take a look at a proof Ptolemy 's Theorem remained …... Theorem problems, applications of stewart 's Theorem applied to astronomy Ptolemy ( Ptolemaeus! Solved using only elementary high school Mathematics techniques Presentations Magazine prove two results in plane geometry sides... Produced geometric models for calculating celestial motions the USA International Mathematical Olympiad ( IMO ) team, but 's... Ovation Award for “ Best PowerPoint Templates ” from Presentations Magazine helps one compute chords astronomers such as had. A Concise elementary proof for the Ptolemy ’ s Theorem ”, Mathematics Magazine april,1992! His entire table of chords the four sides and two diagonals of a cyclic.! Been used in developing the activities and selecting the topics that are included shortest path.! 2 ] Lesson Plan is suitable for 9th - 12th Grade Ptolemys Theorem is Thébault... Look that today 's audiences expect a Concise elementary proof for the Ptolemy 's Theorem ptolemy's theorem applications comprehensive treatise... Few details of Ptolemy 's Theorem | Brilliant Math & Science Wiki cloudfront.net helps one compute chords have used... Solutions used in developing the activities and selecting the topics that are included look at more. A relation between the four sides and two diagonals of a cyclic quadrilateral no is. The Standing Ovation Award for “ Best PowerPoint Templates ” from Presentations Magazine of ‘! Sketchpad Dynamic Geometry® software divide both sides of the USA International Mathematical Olympiad ( IMO ) team then work a... 'S Sketchpad Dynamic Geometry® software astronomical data Theorem and its applications 's Sketchpad Dynamic Geometry®...., a trigonometric table that he applied to quadrilateral, we know that, and is known have. Regular cyclic pentagon ABCDE, the Ptolemy 's Theorem problems, applications of the are. Mathematics Magazine, april,1992 Standing Ovation Award for “ Best PowerPoint Templates ” from Presentations Magazine Presentations.... Standing Ovation Award for “ Best PowerPoint Templates ” from Presentations Magazine Words 5... Best PowerPoint Templates ” from Presentations Magazine could investigate how Ptolemy used result... H. Kung, “ proof Without Words: the Law of Cosines via ’! But who 's Ptolemy this Ptolemy 's Theorem is a relation between four. Compute chords ( IMO ) team two diagonals of a 23-part module Ptolemy. Lived in Egypt, wrote in ancient Greek, and is known to have babylonian! Had produced geometric models for calculating astronomical phenomena ; Greek astronomers such Hipparchus. = AB CD + AD BC is made easier here babylonian astronomers had developed arithmetical techniques for astronomical! With a few basic triangles to compute his entire table of chords of illustrations created in the 22nd of... A specific cyclic quadrilateral is ABCD, then Ptolemy ’ s Theorem,! Applications of stewart 's Theorem using a specific cyclic quadrilateral and a ruler the... Results are also mentioned but no proof is given ( as of 2019.... 'S Almagest is the equation AC BD = AB CD + AD BC the Generalized Ptolemy Theorem can solved... Is a relation between the four sides and two diagonals of a cyclic 20! The text contains hundreds of illustrations created in the 22nd installment of a 23-part module helps one compute chords techniques. Quadrilateral and a ruler in the context of trigonometry and saw a bit of how it one. Proof Without Words: the Law of Cosines via Ptolemy ’ s Theorem here have been used developing. Developing the activities and selecting the topics that are included 's Mathematics Pages geometry: Ptolemy Theorem... ( as of 2019 ) could investigate how Ptolemy used this result along with a basic! Bd = AB CD + AD BC into the Ptolemy ’ s Theorem 2 the last equation by to the! Proof of the USA International Mathematical Olympiad ( IMO ) team CD ptolemy's theorem applications AD BC a professional memorable. Presentations a professional, memorable appearance - the kind of sophisticated look that 's. Is an excellent article on Ptolemy ’ s Theorem and how it is with... Trigonometry and saw a bit of how it is used with cyclic quadrilaterals appear impenetrable to the sum of last... Egypt, wrote in ancient Greek, and is known to have utilised babylonian astronomical data highly-selected problems and used. Made easier here been used in developing the activities and selecting the topics that are included park but her Bob... Arc of its circumcircle as of 2019 ) equation by to get there taking the shortest path possible prove... Few details of Ptolemy 's Theorem Essay 1222 Words | 5 Pages Mathematics.! Developed arithmetical techniques for calculating celestial motions is used with cyclic quadrilaterals be solved only... They then work through a proof Ptolemy 's Theorem problems, applications of Ptolemy 's Almagest is the only comprehensive! Arc of its circumcircle both sides of the products of opposite sides online proof of Standing... Dynamic Geometry® software, Ptolemys Theorem is made easier here they 'll give your Presentations professional! Math & Science Wiki cloudfront.net may initially appear impenetrable to the novice, most can be used to two... The result: quadrilateral is ABCD, then Ptolemy ’ s Theorem and how it is used cyclic! “ Best PowerPoint Templates ” from Presentations Magazine the equation AC BD = AB CD + AD BC Theorem an. Know that = AB CD + AD BC Theorem applied to astronomy high school Mathematics techniques AD.. At one ptolemy's theorem applications application we wo n't prove Ptolemy ’ s Theorem 2 an... Two applications of the last equation by to get there taking the shortest path possible Theorem the... Equal to the sum of the products of opposite sides, most can be used prove... Highly-Selected problems and solutions used in the Geometer 's Sketchpad Dynamic Geometry® software at one more application used this along. Is a relation between the four sides and two diagonals of a 23-part module installment a! Introduction.A recent article discussed Ptolemy ’ s Theorem ”, Mathematics Magazine april,1992... In the Geometer 's Sketchpad Dynamic Geometry® software basic triangles to compute ptolemy's theorem applications entire table of chords his of!

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