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<ADO = 90 ^o wanted, diameter , 2midpoints, perpendiculars related

Source: Mathematics Regional Olympiad of Mexico Center Zone 2009 P1

November 10, 2021
geometryright angle

Problem Statement

Let Γ\Gamma be a circle with the center OO and let AA, A′A ^ \prime be two diametrically opposite points in Γ\Gamma. Let PP be the midpoint of OA′OA ^ \prime and ℓ\ell a line that passes through PP, different from the line AA′AA ^ \prime and different from the line perpendicular on AA′AA ^ \prime. Let BB and CC be the intersection points of ℓ\ell with Γ\Gamma, let HH be the foot of the altitude from AA on BCBC, let MM be the midpoint of BCBC, and let DD be the intersection of the line A′MA ^ \prime M with AHAH. Show that the angle ∠ADO=90∘\angle ADO = 90 ^ \circ .
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