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Reflections of lines through reflections of excenters

Source: 2016 IMO Shortlist G7

July 19, 2017
geometryIMO Shortlist

Problem Statement

Let II be the incentre of a non-equilateral triangle ABCABC, IAI_A be the AA-excentre, IA′I'_A be the reflection of IAI_A in BCBC, and lAl_A be the reflection of line AIA′AI'_A in AIAI. Define points IBI_B, IB′I'_B and line lBl_B analogously. Let PP be the intersection point of lAl_A and lBl_B.
[*] Prove that PP lies on line OIOI where OO is the circumcentre of triangle ABCABC. [*] Let one of the tangents from PP to the incircle of triangle ABCABC meet the circumcircle at points XX and YY. Show that ∠XIY=120∘\angle XIY = 120^{\circ}.
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