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side expression is constant from a line

Source: Mongolia 1999 Grade 10 P2

May 5, 2021
geometry

Problem Statement

The rays l1,l2,…,ln−1l_1,l_2,\ldots,l_{n-1} divide a given angle ABCABC into nn equal parts. A line ll intersects ABAB at A1A_1, BCBC at An+1A_{n+1}, and lil_i at Ai+1A_{i+1} for i=1,…,n−1i=1,\ldots,n-1. Show that the quantity (1BA1+1BAn+1)(1BA1+1BA2+…+1BAn+1)−1\left(\frac1{BA_1}+\frac1{BA_{n+1}}\right)\left(\frac1{BA_1}+\frac1{BA_2}+\ldots+\frac1{BA_{n+1}}\right)^{-1}is independent of the line ll, and compute its value if ∠ABC=ϕ\angle ABC=\phi.
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