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 ${\displaystyle \triangle ABC}$ with a point ${\displaystyle P}$ on its circumcircle the length of longest of the three line segments ⁠${\displaystyle PA}$⁠, ⁠${\displaystyle PB}$⁠, ⁠${\displaystyle PC}$⁠ connecting ${\displaystyle P}$ 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 cyclic quadrilaterals. Let ${\displaystyle a}$ be the side length of the equilateral triangle ${\displaystyle \triangle ABC}$ and ${\displaystyle PA}$ the longest line segment. The triangle's vertices together with ${\displaystyle P}$ form a cyclic quadrilateral ${\displaystyle \square ABPC}$. By Ptolemy's theorem,

${\displaystyle |BC|\cdot |PA|=|AC|\cdot |PB|+|AB|\cdot |PC|.}$

But, since the triangle is equilateral, ${\displaystyle |BC|=|AC|=|AB|=a}$, so

${\displaystyle {\begin{aligned}a\cdot |PA|&=a\cdot |PB|+a\cdot |PC|\\[3mu]|PA|&=|PB|+|PC|.\end{aligned}}}$

• 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

• Van Schooten's theorem at cut-the-knot.org