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

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