Triangles PWZ and OWX are similar; PYZ and OXY are similar, because both triangles are right-angled and angles PWZ and OXW are equal; and angles OXY and PYZ are equal (complementary angles).
OW/OX=PZ/PW; OY/OX=PZ/PY. OW=OX.PZ/PW and PY=OX.PZ/OY.
Therefore, OW/PY=OY/PW=(OP+PY)/(OP+OW)
Cross-multiplying: OW.OP+OW^2=OP.PY+PY^2
Rearranging: OP(OW-PY)+OW^2-PY^2=0=OP(OW-PY)+(OW+PY)(OW-PY)
Thus: (OW+OP+PY)(OW-PY)=0=WY(OW-PY)=0
Since the diagonal WY is non-zero, OW=PY (QED).