Question 23
Let ABCD be a cyclic quadrilateral with circumcircle $ \Omega $ such that $ CD \gt BD \gt BC $. The tangents to $ \Omega $ at B and C meet at a point S. A line $ \ell $ is called coastal if:
• $ \ell $ does not pass through A,
• $ \ell $ intersects ray SD at a point $ X_{\ell} $ beyond D, and
• there is a circle tangent to line $\ell$, line $X_{\ell}A$, segment SB and segment SC.
Prove that there is a circle that is tangent to $ \Omega $ and all coastal lines.
• $ \ell $ does not pass through A,
• $ \ell $ intersects ray SD at a point $ X_{\ell} $ beyond D, and
• there is a circle tangent to line $\ell$, line $X_{\ell}A$, segment SB and segment SC.
Prove that there is a circle that is tangent to $ \Omega $ and all coastal lines.