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if incenter I in (KDE), prove that BD+CE=BC, symmetric lines, perpendicular

Source: 2016 Saudi Arabia GMO TST level 4, II p3

August 1, 2020
geometryincenterperpendicularincircle

Problem Statement

Let ABCABC be a triangle with incenter II . Let CI,BICI, BI intersect AB,ACAB, AC at D,ED, E respectively. Denote by Δb,Δc\Delta_b,\Delta_c the lines symmetric to the lines AB,ACAB, AC with respect to CD,BECD, BE correspondingly. Suppose that Δb,Δc\Delta_b,\Delta_c meet at KK. a) Prove that IK⊥BCIK \perp BC. b) If I∈(KDE)I \in (K DE), prove that BD+CE=BCBD + C E = BC.
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