Triangle Circumcircle-Incentre Result

Given any triangle ABC, construct the angle bisector of angle BAC and extend to meet the circumcircle of ABC in F. Prove that F is the midpoint of arc BC, and if a circle with F as centre and FB as radius is constructed, then the point P, the intersection of this new circle and FA, is the incentre of triangle ABC.

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Triangle Circumcircle-Incentre Result

Can you prove the result?

A proof of the result is given on p. 190 of Some Adventures in Euclidean Geometry, which is available for purchase as downloadable PDF or printed book at More Info

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Michael de Villiers, Dec 2009.