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geometry around incenter and midpoint of arc ACB

Source: Bulgaria NMO 2021 P6

May 16, 2021
geometryincentergeometric transformationreflection

Problem Statement

Point SS is the midpoint of arc ACBACB of the circumscribed circle kk around triangle ABCABC with AC>BCAC>BC. Let II be the incenter of triangle ABCABC. Line SISI intersects kk again at point TT. Let DD be the reflection of II across TT and MM be the midpoint of side ABAB. Line IMIM intersects the line through DD, parallel to ABAB, at point EE. Prove that AE=BDAE=BD.
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