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Circles touches side of triangle

Source: Vietnam TST 1999 for the 40th IMO, problem 5

June 26, 2005
geometryfunctiongeometry solved

Problem Statement

Let a triangle ABCABC inscribed in circle Γ\Gamma be given. Circle Θ\Theta lies in angle A^Â of triangle and touches sides AB,ACAB, AC at M1,N1M_1, N_1 and touches internally Γ\Gamma at P1P_1. The points M2,N2,P2M_2, N_2, P_2 and M3,N3,P3M_3, N_3, P_3 are defined similarly to angles BB and CC respectively. Show that M1N1,M2N2M_1N_1, M_2N_2 and M3N3M_3N_3 intersect each other at their midpoints.
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