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P3 Bosnia and Herzegovina JBMO TST

Source: Bosnia and Herzegovina JBMO TST 2018

July 14, 2018
geometry

Problem Statement

Let Γ\Gamma be circumscribed circle of triangle ABCABC (AB≠AC)(AB \neq AC). Let OO be circumcenter of the triangle ABCABC. Let MM be a point where angle bisector of angle BACBAC intersects Γ\Gamma. Let DD (D≠M)(D \neq M) be a point where circumscribed circle of the triangle BOMBOM intersects line segment AMAM and let EE (E≠M)(E \neq M) be a point where circumscribed circle of triangle COMCOM intersects line segment AMAM. Prove that BD+CE=AMBD+CE=AM.
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