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 2 of 4: Obtain the parallel sides
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∠PRT=∠MPX=∠PFX=∠PRS\angle PRT=\angle MPX=\angle PFX=\angle PRS
Detailed analysis

Because M is the midpoint of the arc AB, the tangent at M to omega is parallel to AB. The tangent-chord theorem gives the displayed chain of equal angles. Hence Q is the midpoint of the arc TS of Omega, so ST is parallel to the tangent at Q, namely ell_Q. Therefore corresponding sides of triangles RST and XYZ are parallel.