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Geometry with incircles

Source: St Petersburg Olympiad 2016, Grade 11, P5

January 24, 2018
geometryincenter

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 I1I2⊥I_1I_2 \perp bisector of ∠ABC\angle ABC
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