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January 4, 2016

Triangle Problem | RMO 2015 Solutions Problem 1

Try the solution of problem from RMO (Regional Mathematical Olympiad) 2015 Problem 1 based on Triangle.

Problem: Triangle Problem

Two circles \Gamma and \Sigma , with centers O and O', respectively, are such that O' lies on \Gamma . Let A be a point on \Sigma , and let M be the midpoint of AO'. Let B be another point on \Sigma , such that AB~||~OM . Then prove that the midpoint of AB lies on \Gamma .



Suppose AB intersects \Sigma at C. Join O'C. Suppose it intersects OM at D. Clearly in \Delta AO'C M is the midpoint of AO' and DM is parallel to AC. Then D is the midpoint of O'C.

Now O'C is a chord of \Gamma and we have proved that D is the midpoint of it. Therefore we can say that OD is perpendicular to O'C.

Since AB is parallel to OD (OM), therefore as O'C is perpendicular to OD, therefore O'C is also perpendicular to AB. Since AB is a chord of circle \Sigma and O'C is a line from center perpendicular to the chord, hence it bisects are chord implying that C is the midpoint of AB.

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