Van Schooten's theorem: A proof with some words
Van Schooten's theorem claims that for an equilateral triangle A1A2A3 with a point P on its circumcircle the length of longest of the three line segments PA1, PA2, PA3 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 PA3 around circumcenter O by 120 degrees counterclockwise.
- Step 3: Rotating the segment PA2 around circumcenter O by 120 degrees counterclockwise.
- Step 4: Measuring angle A1P'A3: it must be 60 degrees because P' lies on the same arc as A2 and they share the same chord A1A2.
- Step 5: Measuring angle A1PA3: it must be 60 degrees because P lies on the same arc as A2 and they share the same chord A1A2.
- Final step: Because of rotation, the supplementary angles at the intersection of chords A1P' and A3P 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.