Van Schooten's theorem explained

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

References

• Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010,, pp. 102–103
• Doug French: Teaching and Learning Geometry. Bloomsbury Publishing, 2004,, 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

External links

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