## 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.