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incenter of a triangle is circumcenter to another, 2 isosceles triangles wanted

Source: KJMO 2007 p7

April 30, 2019
geometryincenterCircumcenterisoscelesIsosceles Triangle

Problem Statement

Let the incircle of △ABC\triangle ABC meet BC,CA,ABBC,CA,AB at J,K,LJ,K,L. Let D(≠B,J),E(≠C,K),F(≠A,L)D(\ne B, J),E(\ne C,K), F(\ne A,L) be points on BJ,CK,ALBJ,CK,AL. If the incenter of △ABC\triangle ABC is the circumcenter of △DEF\triangle DEF and ∠BAC=∠DEF\angle BAC = \angle DEF, prove that △ABC\triangle ABC and △DEF\triangle DEF are isosceles triangles.
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