# Van Schooten's theorem: A proof with some words

Van Schooten's theorem claims that for an equilateral triangle

*A*_{1}*A*_{2}*A*_{3}with a point*P*on its circumcircle the length of longest of the three line segments*PA*_{1},*PA*_{2},*PA*_{3}connecting*P*with the vertices of the triangle equals the sum of the lengths of the other two. This proof is based on Raymond Viglione's proof without words (Mathematics Magazine, Vol. 89, No. 2 (April 2016), p. 132), but here the steps are expressed in a more detailed way:- Step 1: Problem setting.
Step 2: Rotating the segment
*PA*_{3}around circumcenter*O*by 120 degrees counterclockwise. - Step 3: Rotating the segment
*PA*_{2}around circumcenter*O*by 120 degrees counterclockwise. - Step 4: Measuring angle
*A*_{1}*P*'*A*_{3}: it must be 60 degrees because P' lies on the same arc as*A*_{2}and they share the same chord*A*_{1}*A*_{2}. - Step 5: Measuring angle
*A*_{1}*PA*_{3}: it must be 60 degrees because P lies on the same arc as*A*_{2}and they share the same chord*A*_{1}*A*_{2}. - Final step: Because of rotation, the supplementary angles at the intersection of chords
*A*_{1}*P*' and*A*_{3}*P*are 60 degrees. So are the red and green triangles regular, both have the same side lengths, and the length*c*is clearly the sum of*a*and*b*.

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