Van Schooten's theorem

Short description: On lines connecting the vertices of an equilateral triangle to a point on its circumcircle

[math]\displaystyle{ |PA| = |PB| + |PC| }[/math]

Van Schooten's theorem, named after the Dutch mathematician Frans van Schooten, describes a property of equilateral triangles. It states:

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

[math]\displaystyle{ \begin{align} & |BC| \cdot |PA| =|AC| \cdot |PB| + |AB| \cdot |PC| \\[6pt] \Longleftrightarrow & a \cdot |PA| =a \cdot |PB| + a \cdot |PC| \end{align} }[/math]

Dividing the last equation by [math]\displaystyle{ a }[/math] 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