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Geometry with incircles II

Source: St Petersburg Olympiad 2016, Grade 9, P6

September 8, 2018
geometryincircleincenterPetersburgGrade 9

Problem Statement

Incircle of △ABC\triangle ABC touch ACAC at DD. BDBD intersect incircle at EE. Points F,GF,G on incircle are such points, that FE∥BC,GE∥ABFE \parallel BC,GE \parallel AB. I1,I2I_1,I_2 are incenters of DEF,DEGDEF,DEG. Prove that angle bisector of ∠GDF\angle GDF passes though the midpoint of I1I2I_1I_2 .
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