For an equilateral triangle with a point on its circumcircle the length of longest of the three line segments connecting with the vertices of the triangle equals the sum of the lengths of the other two.
The theorem is a consequence of Ptolemy's theorem for concyclic quadrilaterals. Let be the side length of the equilateral triangle and the longest line segment. The triangle's vertices together with form a concyclic quadrilateral and hence Ptolemy's theorem yields:
Dividing the last equation by delivers Van Schooten's theorem.
ReferencesEdit
Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, ISBN9780883853481, pp. 102–103
Doug French: Teaching and Learning Geometry. Bloomsbury Publishing, 2004, ISBN9780826434173 , pp. 62–64
Raymond Viglione: Proof Without Words: van Schooten′s Theorem. Mathematics Magazine, Vol. 89, No. 2 (April 2016), p. 132
Jozsef Sandor: On the Geometry of Equilateral Triangles. Forum Geometricorum, Volume 5 (2005), pp. 107–117
External linksEdit
Wikimedia Commons has media related to Van Schooten's theorem.
Van Schooten's theorem at cut-the-knot.org
October 21, 2023
schooten, theorem, named, after, dutch, mathematician, frans, schooten, describes, property, equilateral, triangles, states, displaystyle, equilateral, triangle, displaystyle, triangle, with, point, displaystyle, circumcircle, length, longest, three, line, seg. Van Schooten s theorem named after the Dutch mathematician Frans van Schooten describes a property of equilateral triangles It states P A P B P C displaystyle PA PB PC For an equilateral triangle A B C displaystyle triangle ABC with a point P displaystyle P on its circumcircle the length of longest of the three line segments P A P B P C displaystyle PA PB PC connecting P displaystyle P with the vertices of the triangle equals the sum of the lengths of the other two The theorem is a consequence of Ptolemy s theorem for concyclic quadrilaterals Let a displaystyle a be the side length of the equilateral triangle A B C displaystyle triangle ABC and P A displaystyle PA the longest line segment The triangle s vertices together with P displaystyle P form a concyclic quadrilateral and hence Ptolemy s theorem yields B C P A A C P B A B P C a P A a P B a P C displaystyle begin aligned amp BC cdot PA AC cdot PB AB cdot PC 6pt Longleftrightarrow amp a cdot PA a cdot PB a cdot PC end aligned Dividing the last equation by a displaystyle a delivers Van Schooten s theorem References EditClaudi Alsina Roger B Nelsen Charming Proofs A Journey Into Elegant Mathematics MAA 2010 ISBN 9780883853481 pp 102 103 Doug French Teaching and Learning Geometry Bloomsbury Publishing 2004 ISBN 9780826434173 pp 62 64 Raymond Viglione Proof Without Words van Schooten s Theorem Mathematics Magazine Vol 89 No 2 April 2016 p 132 Jozsef Sandor On the Geometry of Equilateral Triangles Forum Geometricorum Volume 5 2005 pp 107 117External links Edit nbsp Wikimedia Commons has media related to Van Schooten s theorem Van Schooten s theorem at cut the knot org Retrieved from https en wikipedia org w index php title Van Schooten 27s theorem amp oldid 1144552210, wikipedia, wiki, book, books, library,
