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one of concstructed circles is tangent to circumcircle

Source: - All-Russian MO 2003 Regional (R4) 10.6

September 17, 2024
geometrytangent circlescircumcircle

Problem Statement

Let A0A_0 be the midpoint of side BCBC of triangle ABCABC, and A′A' be the point of tangency with this side of the inscribed circle. Let's construct a circle ω \omega with center at A0A_0 and passing through A′A'. On other sides we will construct similar circles. Prove that if ω \omega is tangent to the cirucmscribed circle on arc BCBC not containing AA, then another one of the constructed circles touches the circumcircle.
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