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Show that a circumcircle is tangent to Omega

Source: APMO 2014 Problem 5

March 28, 2014
geometrycircumcircleAPMO

Problem Statement

Circles ω\omega and Ω\Omega meet at points AA and BB. Let MM be the midpoint of the arc ABAB of circle ω\omega (MM lies inside Ω\Omega). A chord MPMP of circle ω\omega intersects Ω\Omega at QQ (QQ lies inside ω\omega). Let ℓP\ell_P be the tangent line to ω\omega at PP, and let ℓQ\ell_Q be the tangent line to Ω\Omega at QQ. Prove that the circumcircle of the triangle formed by the lines ℓP\ell_P, ℓQ\ell_Q and ABAB is tangent to Ω\Omega.
Ilya Bogdanov, Russia and Medeubek Kungozhin, Kazakhstan
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