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Geometry Difficulty 7.0 National Olympiad, round 2 Prove it United Kingdom

Let $ABC$ be an acute-angled triangle with $AB > AC$ . Let $M$ be the midpoint of $BC$ . The circle passing through $M$ that is tangent to $AB$ at $B$ and the circle passing through $M$ that is tangent to $AC$ at $C$ intersect again at $D$ . Prove that $MA \times MD = MB \times MC$ .

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