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
& , BC, \cdot , PA, =, AC, \cdot , PB, + , AB, \cdot , PC, \\ pt \Longleftrightarrow & a \cdot , PA, =a \cdot , PB, + a \cdot , PC,
\end

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.&nbs
102–103
*Doug French: ''Teaching and Learning Geometry''. Bloomsbury Publishing, 2004, , pp.&nbs
62–64
*Raymond Viglione: ''Proof Without Words: van Schooten′s Theorem''. Mathematics Magazine, Vol. 89, No. 2 (April 2016),
 132
*Jozsef Sandor
''On the Geometry of Equilateral Triangles''
Forum Geometricorum, Volume 5 (2005), pp. 107–117

External links

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

Euclidean geometry
Theorems about triangles and circles
Theorems about equilateral triangles