MathDB
A Familiar Point

Source: EGMO 2023/6

April 16, 2023
EGMOgeometryEGMO 2023

Problem Statement

Let ABCABC be a triangle with circumcircle Ω\Omega. Let SbS_b and ScS_c respectively denote the midpoints of the arcs ACAC and ABAB that do not contain the third vertex. Let NaN_a denote the midpoint of arc BACBAC (the arc BCBC including AA). Let II be the incenter of ABCABC. Let ωb\omega_b be the circle that is tangent to ABAB and internally tangent to Ω\Omega at SbS_b, and let ωc\omega_c be the circle that is tangent to ACAC and internally tangent to Ω\Omega at ScS_c. Show that the line INaIN_a, and the lines through the intersections of ωb\omega_b and ωc\omega_c, meet on Ω\Omega.
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