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Part of 2017 Oral Moscow Geometry Olympiad

Problems(2)

smallest radius of the circumcircle

Source: 2017 Oral Moscow Geometry Olympiad grades 8-9 p4

7/26/2019
We consider triangles ABCABC, in which the point MM lies on the side ABAB, AM=aAM = a, BM=bBM = b, CM=cCM = c (c<a,c<bc <a, c <b). Find the smallest radius of the circumcircle of such triangles.
geometrycircumcirclecircumradiusradiusminimum value
circle tangent to the incircle criterion using radius of excircle

Source: 2017 Oral Moscow Geometry Olympiad grades 10-11 p4

7/25/2019
Prove that a circle constructed with the side ABAB of a triangle ABCABC as a diameter touches the inscribed circle of the triangle ABCABC if and only if the side ABAB is equal to the radius of the exircle on that side.
geometryincircleexcircleexradiustangent circles