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tangent circle to 3 given circles internally also tangent to a 4th circle

Source: 2016 Saudi Arabia Pre-TST Level 4+ 1.1

September 13, 2020
geometrytangent circlesaltitudes

Problem Statement

Let ABCABC be an acute, non isosceles triangle, AX,BY,CZAX, BY, CZ are the altitudes with X,Y,ZX, Y, Z belong to BC,CA,ABBC, CA,AB respectively. Respectively denote (O1),(O2),(O3)(O_1), (O_2), (O_3) as the circumcircles of triangles AYZ,BZX,CXYAY Z, BZX, CX Y . Suppose that (K)(K) is a circle that internal tangent to (O1),(O2),(O3)(O_1), (O_2), (O_3). Prove that (K)(K) is tangent to circumcircle of triangle ABCABC.
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