The bisector of angle $\displaystyle A $ of the triangle $\displaystyle ABC $ meets the opposite side at a point $\displaystyle P.$ If $\displaystyle Q $ and $\displaystyle R $ are the respective feet from $\displaystyle B $ to $\displaystyle AP $ and from $\displaystyle C $ to $\displaystyle AP $, prove that the points $\displaystyle A,P,Q $and $\displaystyle R $ are harmonic.