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Mongolian Mathematical Olympiad 46

Geometry Difficulty 5.0 AIME Prove it Mongolia

A circle touches $BC$ side of triangle $ABC$ at the point $M$ and intersects sides $AB$ and $AC$ at $D$ and $E$ respectively. If $DE \parallel BC$ , prove that $EM = MD$ .

Solutions — 2

Solution 1

Since $BC$ is tangent to the circle, we have $\angle EMC = \angle EAM = \angle EDM$ and $\angle DMB = \angle DAM = \angle DEM$ . From $DE \parallel BC$ , we have $\angle EDM = \angle DMB$ . So $AM$ bisects $\angle CAB$ .

Solution 2

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