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 ${\displaystyle PA,PB,PC}$ 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 concyclic 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 concyclic quadrilateral and hence Ptolemy's theorem yields:

${\displaystyle {\begin{aligned}&|BC|\cdot |PA|=|AC|\cdot |PB|+|AB|\cdot |PC|\\[6pt]\Longleftrightarrow &a\cdot |PA|=a\cdot |PB|+a\cdot |PC|\end{aligned}}}$

Dividing the last equation by ${\displaystyle a}$ delivers Van Schooten's theorem.