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Concurrency in a quadrilateral implies equal length

Source: 2022 China TST, Test 4 P2

April 30, 2022
geometryincenterangle bisector

Problem Statement

Let ABCDABCD be a convex quadrilateral, the incenters of △ABC\triangle ABC and △ADC\triangle ADC are I,JI,J, respectively. It is known that AC,BD,IJAC,BD,IJ concurrent at a point PP. The line perpendicular to BDBD through PP intersects with the outer angle bisector of ∠BAD\angle BAD and the outer angle bisector ∠BCD\angle BCD at E,FE,F, respectively. Show that PE=PFPE=PF.
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