In triangle ABC, angle A = 40°. The internal bisector of angle B and the external bisector of the angle at C meet at the point P. What is the measure of angle BPC?

The external bisector at C makes angle BCP = C + (180° − C)/2 = 90° + C/2. In triangle BPC, angle BPC = 180° − B/2 − (90° + C/2) = 90° − (B + C)/2 = 90° − (180° − A)/2 = A/2. So angle BPC = 40°/2 = 20°.