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IMOC 2017 G7 (concyclic wanted, <ABD =< DCA amd circumcircle related)

Source: https://artofproblemsolving.com/community/c6h1740077p11309077

March 20, 2020
geometrycircumcircleConcyclicequal angles

Problem Statement

Given △ABC\vartriangle ABC with circumcenter OO. Let DD be a point satisfying ∠ABD=∠DCA\angle ABD = \angle DCA and MM be the midpoint of ADAD. Suppose that BM,CMBM,CM intersect circle (O)(O) at another points E,FE, F, respectively. Let PP be a point on EFEF so that APAP is tangent to circle (O)(O). Prove that A,P,M,OA, P,M,O are concyclic. https://2.bp.blogspot.com/-gSgUG6oywAU/XnSKTnH1yqI/AAAAAAAALdw/3NuPFuouCUMO_6KbydE-KIt6gCJ4OgWdACK4BGAYYCw/s320/imoc2017%2Bg7.png
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