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Circles in trapezoid

Source: China TST 2006 (1)

March 24, 2006
geometrytrapezoidincentergeometric transformationhomothetyratiogeometry unsolved

Problem Statement

ABCDABCD is a trapezoid with AB∣∣CDAB || CD. There are two circles ω1\omega_1 and ω2\omega_2 is the trapezoid such that ω1\omega_1 is tangent to DADA, ABAB, BCBC and ω2\omega_2 is tangent to BCBC, CDCD, DADA. Let l1l_1 be a line passing through AA and tangent to ω2\omega_2(other than ADAD), Let l2l_2 be a line passing through CC and tangent to ω1\omega_1 (other than CBCB).
Prove that l1∣∣l2l_1 || l_2.
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