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BLI , L'IB are similar iff AC = AB + BL , incenter, circumcircle related

Source: Mathematics Regional Olympiad of Mexico Center Zone 2008 P2

November 9, 2021
geometrysimilar triangles

Problem Statement

Let ABCABC be a triangle with incenter II , the line AIAI intersects BCBC at L L and the circumcircle of ABCABC at L′L'. Show that the triangles BLIBLI and L′IBL'IB are similar if and only if AC=AB+BLAC = AB + BL.
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