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Another circle tangent to circle problem

Source: INAMO 2023 P7 (OSN 2023)

August 30, 2023
geometrytangentcircumcircleIndonesiaIndonesia MO

Problem Statement

Given a triangle ABCABC with ∠ACB=90∘\angle ACB = 90^{\circ}. Let ω\omega be the circumcircle of triangle ABCABC. The tangents of ω\omega at BB and CC intersect at PP. Let MM be the midpoint of PBPB. Line CMCM intersects ω\omega at NN and line PNPN intersects ABAB at EE. Point DD is on CMCM such that ED∥BMED \parallel BM. Show that the circumcircle of CDECDE is tangent to ω\omega.
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