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Source: 2024 Vietnam National Olympiad - Problem 3

January 5, 2024
geometry

Problem Statement

Let ABCABC be an acute triangle with circumcenter OO. Let A′A' be the center of the circle passing through CC and tangent to ABAB at AA, let B′B' be the center of the circle passing through AA and tangent to BCBC at BB, let C′C' be the center of the circle passing through BB and tangent to CACA at CC.
a) Prove that the area of triangle A′B′C′A'B'C' is not less than the area of triangle ABCABC. b) Let X,Y,ZX, Y, Z be the projections of OO onto lines A′B′,B′C′,C′A′A'B', B'C', C'A'. Given that the circumcircle of triangle XYZXYZ intersects lines A′B′,B′C′,C′A′A'B', B'C', C'A' again at X′,Y′,Z′X', Y', Z' (X′≠X,Y′≠Y,Z′≠ZX' \neq X, Y' \neq Y, Z' \neq Z), prove that lines AX′,BY′,CZ′AX', BY', CZ' are concurrent.
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