How can we find a point A on OX and a point B on OY such that P is the midpoint of a segment AB drawn between them?

Construct a circle with P as center and
passing through point O. The ray OP 
intersects the circle again at point C.
Construct a line through point C and
parallel to line OY which intersects
line OX at point A.
Construct a line through point C and
parallel to line OX which intersects
line OY at point B.
Point P is the midpoint of diagonal OC
of parallogram AOBC. Therefore, point P
is the midpoint of diagonal AB.
QED