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2 circles tangent internally to circumcircle, inscribed in <ADC, <BDC

Source: Sharygin 2011 Final 10.4

March 31, 2019
geometrycircumcircletangent circlesperpendicularincenter

Problem Statement

Point DD lies on the side ABAB of triangle ABCABC. The circle inscribed in angle ADCADC touches internally the circumcircle of triangle ACDACD. Another circle inscribed in angle BDCBDC touches internally the circumcircle of triangle BCDBCD. These two circles touch segment CDCD in the same point XX. Prove that the perpendicular from XX to ABAB passes through the incenter of triangle ABCABC
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