On lines connecting the vertices of an equilateral triangle to a point on its circumcircle
Van Schooten's theorem, named after the Dutch mathematician Frans van Schooten, describes a property of equilateral triangles. It states:
"For an equilateral triangle formula_1 with a point formula_2 on its circumcircle the length of longest of the three line segments formula_3 connecting formula_2 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 formula_5 be the side length of the equilateral triangle formula_1 and formula_7 the longest line segment. The triangle's vertices together with formula_2 form a concyclic quadrilateral and hence Ptolemy's theorem yields:
formula_9
Dividing the last equation by formula_5 delivers Van Schooten's theorem.