[11 points] Given $\triangle XYZ$, the following constructions are made: mark point $W$ on segment $XZ$, point $P$ on segment $XW$ and point $Q$ on segment $YZ$ such that
$$\frac{WZ}{YX} = \frac{PW}{XP} = \frac{QZ}{YQ} = k$$
Extend segments $QP$ and $YX$ to meet at the point $R$ as shown. Prove that $XR = XP$.
Hint (use this or your own method): A suitable construction may help in calculations.