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D circumcenter of MNK wanted, < CAP = < CBP = <ACB

Source: 2021 3nd Final Mathematical Cup Junior Division P2 FMC

October 13, 2021
geometryCircumcenterequal angles

Problem Statement

Let ABCABC be an acute triangle, where ABAB is the smallest side and let DD be the midpoint of ABAB. Let PP be a point in the interior of the triangle ABCABC such that ∠CAP=∠CBP=∠ACB\angle CAP = \angle CBP = \angle ACB. From the point PP, we draw perpendicular lines on BCBC and ACAC where the intersection point with BCBC is MM, and with ACAC is NN . Through the point MM we draw a line parallel to ACAC, and through NN parallel to BCBC. These lines intercept at the point KK. Prove that DD is the center of the circumscribed circle for the triangle MNKMNK.
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