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Part of 2011 Israel National Olympiad

Problems(1)

Israel 2011 Q4 - Four tangent circles

Source: Israel National Olympiad 2011 Q4

8/8/2019
Let α1,α2,α3\alpha_1,\alpha_2,\alpha_3 be three congruent circles that are tangent to each other. A third circle β\beta is tangent to them at points A1,A2,A3A_1,A_2,A_3 respectively. Let PP be a point on β\beta which is different from A1,A2,A3A_1,A_2,A_3. For i=1,2,3i=1,2,3, let BiB_i be the second intersection point of the line PAiPA_i with circle αi\alpha_i. Prove that ΔB1B2B3\Delta B_1B_2B_3 is equilateral.
geometrytangent circlesEquilateral Triangle