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.
The theorem is a consequence of Ptolemy's theorem for cyclic quadrilaterals. Let be the side length of the equilateral triangle and the longest line segment. The triangle's vertices together with form a cyclic quadrilateral By Ptolemy's theorem,
But, since the triangle is equilateral, so
References
- Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, , pp. 102âÂÂ103
- Doug French: Teaching and Learning Geometry. Bloomsbury Publishing, 2004, , pp. 62âÂÂ64
- 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. 107âÂÂ117
External links