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A sequence of circles

Source: Indian Postal Coaching 2005

September 22, 2005
geometry unsolvedgeometry

Problem Statement

Let <Γj>< \Gamma _j > be a sequnce of concentric circles such that the sequence <Rj>< R_j > , where RjR_j denotes the radius of Γj\Gamma_j, is increasing and Rj⟶∞R_j \longrightarrow \infty as j⟶∞j \longrightarrow \infty. Let A1B1C1A_1 B_1 C_1 be a triangle inscribed in Γ1\Gamma _1. extend the rays AiB1⃗,B1C1⃗,C1A1⃗\vec{A_i B_1} , \vec{B_1 C_1 }, \vec{C_1 A_1} to meet Γ2\Gamma_2 in B2,C2B_2, C_2and A2A_2 respectively and form the triangle A2B2C2A_2 B_2 C_2. Continue this process. Show that the sequence of triangles <AnBnCn>< A_n B_n C_n > tends to an equilateral triangle as n⟶∞n \longrightarrow \infty
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