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Incenter related geometry with specific ratio

Source: 2022 Bulgarian Spring Math Competition, Problem 10.2

March 27, 2022
geometryincenterarc midpointratio

Problem Statement

Let △ABC\triangle ABC have incenter II. The line CICI intersects the circumcircle of △ABC\triangle ABC for the second time at LL, and CI=2ILCI=2IL. Points MM and NN lie on the segment ABAB, such that ∠AIM=∠BIN=90∘\angle AIM =\angle BIN = 90^{\circ}. Prove that AB=2MNAB=2MN.
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