IL-GILLIS 2021 Problem 7

Triangle $ABC$ is given. The circle $\omega$ with center $I$ is tangent at points $D,E,F$ to segments $BC,AC,AB$ respectively. When $ABC$ is rotated $180$ degrees about point $I$ , triangle $A'B'C'$ results. Lines $AD, B'C'$ meet at $U$ , lines $BE, A'C'$ meet at $V$ , and lines $CF, A'B'$ meet at $W$ . Line $BC$ meets $A'C', A'B'$ at points $D_1, D_2$ respectively. Line $AC$ meets $A'B', B'C'$ at $E_1, E_2$ respectively. Line $AB$ meets $B'C', A'C'$ at $F_1,F_2$ respectively. Six (not necessarily convex) quadrilaterals were colored orange: $$AUIF_2 , C'FIF_2 , BVID_1 , A'DID_2 , CWIE_1 , B'EIE_2$$ Six other quadrilaterals were colored green: $$AUIE_2 , C'FIF_1 , BVIF_2 , A'DID_1 , CWID_2 , B'EIE_1$$ Prove that the sum of the green areas equals the sum of the orange areas.

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