Van Schooten's theorem
Van Schooten's theorem, named after the Dutch mathematician Frans van Schooten, describes a property of equilateral triangles. It states:
- 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.
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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.
References
- Claudi 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. 107–117
External links
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Wikimedia Commons has media related to Van Schooten's theorem.
- Van Schooten's theorem at cut-the-knot.org
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