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3

Part of 2006 Oral Moscow Geometry Olympiad

Problems(2)

A'B' > diameter of incircle, <A'C'B'=90^o

Source: 2006 Oral Moscow Geometry Olympiad grades 8-9 p3

10/17/2020
On the sides AB,BCAB, BC and ACAC of the triangle ABCABC, points C′,A′C', A' and B′B' are selected, respectively, so that the angle A′C′B′A'C'B' is right. Prove that the segment A′B′A'B' is longer than the diameter of the inscribed circle of the triangle ABCABC.
(M. Volchkevich)
geometrygeometric inequalityinradiusright angle
locus of centroids of triangles, 2 vertices on one circle, 3rd on another

Source: 2006 Oral Moscow Geometry Olympiad grades 10-11 p3

10/20/2020
Two non-rolling circles C1C_1 and C2C_2 with centers O1O_1 and O2O_2 and radii 2R2R and RR, respectively, are given on the plane. Find the locus of the centers of gravity of triangles in which one vertex lies on C1C_1 and the other two lie on C2C_2.
(B. Frenkin)
geometryLocusCentroid