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midpoints equivalency and a tangent ?

Source: Mexico National Olympiad 2021 Problem 2

November 10, 2021
geometry

Problem Statement

Let ABCABC be a triangle with ∠ACB>90∘\angle ACB > 90^{\circ}, and let DD be a point on BCBC such that ADAD is perpendicular to BCBC. Consider the circumference Γ\Gamma with with diameter BCBC. A line ℓ\ell passes through DD and is tangent to Γ\Gamma at PP, cuts ACAC at MM (such that MM is in between AA and CC), and cuts the side ABAB at NN. Prove that MM is the midpoint of DPDP if and only if NN is the midpoint of ABAB.
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