In triangle ABC the angle at A measures 80°. The bisectors of the exterior angles at B and at C meet at the point P. What is the measure of ∠BPC?

The exterior angle at B is 180° − B, so its bisector makes ∠PBC = 90° − B/2, and likewise ∠PCB = 90° − C/2. In triangle BPC the angles add to 180°, so ∠BPC = 180° − (90° − B/2) − (90° − C/2) = (B + C)/2 = (180° − 80°)/2 = 50°. That is the standard excentre result ∠BPC = 90° − A/2. The trap 130° is 90° + A/2, which is the formula for the INCENTRE, where the internal bisectors meet, and 40° is just A/2.