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Two externally tangent circles and two tangent paralel lines

Source: Romanian IMO Team Selection Test TST 2003, problem 8

September 24, 2005
geometrycircumcircleinvariantgeometry proposed

Problem Statement

Two circles ω1\omega_1 and ω2\omega_2 with radii r1r_1 and r2r_2, r2>r1r_2>r_1, are externally tangent. The line t1t_1 is tangent to the circles ω1\omega_1 and ω2\omega_2 at points AA and DD respectively. The parallel line t2t_2 to the line t1t_1 is tangent to the circle ω1\omega_1 and intersects the circle ω2\omega_2 at points EE and FF. The line t3t_3 passing through DD intersects the line t2t_2 and the circle ω2\omega_2 in BB and CC respectively, both different of EE and FF respectively. Prove that the circumcircle of the triangle ABCABC is tangent to the line t1t_1. Dinu Serbanescu
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