Question 21
Let ABCD be a convex quadrilateral with $ \angle ADC \gt 90^{\circ} $. Suppose that points X and Y lie on diagonal AC such that
$$ \angle ADX=\angle YDC=90^{\circ},\quad\angle CAD=\angle CBX,\quad \text{and}\quad\angle DCA=\angle YBA. $$
Let O be the circumcentre of triangle BXY. Suppose point $ R \neq O $ lies on the perpendicular bisector of AC such that lines OR and AC are parallel.
Prove that line RO bisects angle $ \angle DRB $.
$$ \angle ADX=\angle YDC=90^{\circ},\quad\angle CAD=\angle CBX,\quad \text{and}\quad\angle DCA=\angle YBA. $$
Let O be the circumcentre of triangle BXY. Suppose point $ R \neq O $ lies on the perpendicular bisector of AC such that lines OR and AC are parallel.
Prove that line RO bisects angle $ \angle DRB $.