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{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 $a$ delivers Van Schooten's theorem.

References

Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, ISBN 9780883853481, pp. 102–103
Doug French: Teaching and Learning Geometry. Bloomsbury Publishing, 2004, ISBN 9780826434173 , 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