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 $$\triangle ABC$$ with a point $$P$$ on its circumcircle the length of longest of the three line segments $$PA, PB, PC$$ connecting $$P$$ 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 $$a$$ be the side length of the equilateral triangle $$\triangle ABC$$ and $$PA$$ the longest line segment. The triangle's vertices together with $$P$$ form a concyclic quadrilateral and hence Ptolemy's theorem yields:

\begin{align} & |BC| \cdot |PA| =|AC| \cdot |PB| + |AB| \cdot |PC| \\[6pt] \Longleftrightarrow & a \cdot |PA| =a \cdot |PB| + a \cdot |PC| \end{align} $$ Dividing the last equation by $$a$$ delivers Van Schooten's theorem.