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Concurrent Lines - Incenter and Circumcircles

Source: 2015 Korean Junior MO P5

November 1, 2015
geometryincentercircumcircle

Problem Statement

Let II be the incenter of an acute triangle △ABC\triangle ABC, and let the incircle be Γ\Gamma. Let the circumcircle of △IBC\triangle IBC hit Γ\Gamma at D,ED, E, where DD is closer to BB and EE is closer to CC. Let Γ∩BE=K(≠E)\Gamma \cap BE = K (\not= E), CD∩BI=TCD \cap BI = T, and CD∩Γ=L(≠D)CD \cap \Gamma = L (\not= D). Let the line passing TT and perpendicular to BIBI meet Γ\Gamma at PP, where PP is inside △IBC\triangle IBC. Prove that the tangent to Γ\Gamma at PP, KLKL, BIBI are concurrent.
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