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18

Part of 2010 Sharygin Geometry Olympiad

Problems(1)

Prove that FG passes through the midpoint of AC (18)

Source:

10/29/2010
A point BB lies on a chord ACAC of circle ω.\omega. Segments ABAB and BCBC are diameters of circles ω1\omega_1 and ω2\omega_2 centered at O1O_1 and O2O_2 respectively. These circles intersect ω\omega for the second time in points DD and EE respectively. The rays O1DO_1D and O2EO_2E meet in a point F,F, and the rays ADAD and CECE do in a point G.G. Prove that the line FGFG passes through the midpoint of the segment AC.AC.
geometryparallelogramincentergeometric transformationhomothetyprojective geometrygeometry unsolved