Mongolian National Mathematical Olympiad

Question

Let ABCD be a cyclic quadrilateral with circumcenter , and E be the intersection of the diagonals AC and BD. A line passing through E intersects lines AB, BC at P, Q, respectively. Let R (R D) be the intersection point of and a circle that passes through D, E and tangents the line PQ at E. Prove that B, P, Q, R are cyclic. (Proposed by B. Khoroldagva) FIGURE_0

Accepted Answer

Let _1 be the circle that passes through D, E and tangents the line PQ at E. Since PQ is tangent to the circle _1 at E we have EDR = QER. So it follows from EDR = BAR that PAR = PER, i.e, P, A, E, R are cyclic. Thus, since RBC = RAE we get RPQ = RBQ and so P, B, Q, R are cyclic. FIGURE_0