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(CEG) tangent to AC

Source: 2022 Greece JBMO TST p2

November 3, 2022
geometrytangent

Problem Statement

Let ABCABC be an acute triangle with AB<AC<BCAB<AC < BC, inscirbed in circle Γ1\Gamma_1, with center OO. Circle Γ2\Gamma_2, with center point AA and radius ACAC intersects BCBC at point DD and the circle Γ1\Gamma_1 at point EE. Line ADAD intersects circle Γ1\Gamma_1 at point FF. The circumscribed circle Γ3\Gamma_3 of triangle DEFDEF, intersects BCBC at point GG. Prove that: a) Point BB is the center of circle Γ3\Gamma_3 b) Circumscribed circle of triangle CEGCEG is tangent to ACAC.
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