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geometry iff again

Source: Indonesia IMO 2007 TST, Stage 2, Test 4, Problem 2

November 15, 2009
geometrygeometry proposed

Problem Statement

Let ABCD ABCD be a convex quadrtilateral such that AB AB is not parallel with CD CD. Let Γ1 \Gamma_1 be a circle that passes through A A and B B and is tangent to CD CD at P P. Also, let Γ2 \Gamma_2 be a circle that passes through C C and D D and is tangent to AB AB at Q Q. Let the circles Γ1 \Gamma_1 and Γ2 \Gamma_2 intersect at E E and F F. Prove that EF EF passes through the midpoint of PQ PQ iff BC∥AD BC \parallel AD.
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