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 4 of 4: Conclude tangency
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D∈Ω⟹(XYZ) and (RST) are tangentD\in\Omega\Longrightarrow (XYZ)\text{ and }(RST)\text{ are tangent}
Detailed analysis

The two triangles are related by a homothety centered at D, and D lies on Omega. Their circumcircles are therefore tangent at D; the circumcircle of XYZ is exactly the circle in the statement. This proves the claim.