MathDB
China 2010 quiz4 problem 1

Source:

September 13, 2010
geometrycircumcircleratiosimilar trianglesgeometry unsolved

Problem Statement

Let △ABC\triangle ABC be an acute triangle, and let DD be the projection of AA on BCBC. Let M,NM,N be the midpoints of ABAB and ACAC respectively. Let Γ1\Gamma_1 and Γ2\Gamma_2 be the circumcircles of △BDM\triangle BDM and △CDN\triangle CDN respectively, and let KK be the other intersection point of Γ1\Gamma_1 and Γ2\Gamma_2. Let PP be an arbitrary point on BCBC and E,FE,F are on ACAC and ABAB respectively such that PEAFPEAF is a parallelogram. Prove that if MNMN is a common tangent line of Γ1\Gamma_1 and Γ2\Gamma_2, then K,E,A,FK,E,A,F are concyclic.
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