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The cricles $\Gamma_1$ and $\Gamma_2$ intersect in $M$ and $N$.

Source: Moldova TST 2004

March 8, 2023
geometry

Problem Statement

The cricles Γ1\Gamma_1 and Γ2\Gamma_2 intersect in M M and N.N. A line that goes through M M intersects the cricles Γ1\Gamma_1 and Γ2\Gamma_2 in A A and BB, such that M∈(AB)M\in(AB). The bisector of angle AMN AMN intersects the circle Γ1\Gamma_1 in D,D, and the bisector of angle BMNBMN intersects the circle Γ2\Gamma_2 in C.C. Prove that the circle with diameter CDCD splits the segment ABAB in half.
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