MathLabs

Problem 5

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.
Step 1 of 4: Name the line intersections
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X=AB∩ℓP,Y=AB∩ℓQ,Z=ℓP∩ℓQ,F=MP∩ABX=AB\cap\ell_P,\quad Y=AB\cap\ell_Q,\quad Z=\ell_P\cap\ell_Q,\quad F=MP\cap AB
Detailed analysis

Let X, Y, and Z be the three pairwise intersections of the given lines, and let F be the intersection of MP with AB. Let R be the second intersection of PQ with Omega; let S be the point of Omega with SR parallel to AB; and let T be the point of Omega with RT parallel to the tangent at P.