MathDB
Congruent angles and Miquel's point !!

Source: Romania TST 2015 Day 5 Problem 1

June 4, 2015
Miquel pointequal anglesgeometryRomanian TSTBritishMathematicalOlympiadSpiral Similarity

Problem Statement

Let ABCABC be a triangle. Let P1P_1 and P2P_2 be points on the side ABAB such that P2P_2 lies on the segment BP1BP_1 and AP1=BP2AP_1 = BP_2; similarly, let Q1Q_1 and Q2Q_2 be points on the side BCBC such that Q2Q_2 lies on the segment BQ1BQ_1 and BQ1=CQ2BQ_1 = CQ_2. The segments P1Q2P_1Q_2 and P2Q1P_2Q_1 meet at RR, and the circles P1P2RP_1P_2R and Q1Q2RQ_1Q_2R meet again at SS, situated inside triangle P1Q1RP_1Q_1R. Finally, let MM be the midpoint of the side ACAC. Prove that the angles P1RSP_1RS and Q1RMQ_1RM are equal.
Back to ProblemsView on AoPS