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Prove existence of points on sides of triangle

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October 12, 2010
geometry unsolvedgeometry

Problem Statement

Let cc be the inscribed circle of the triangle ABCABC, dd a line tangent to cc which does not pass through the vertices of triangle ABCABC. Prove the existence of points A1,B1,C1A_1,B_1, C_1, respectively, on the lines BC,CA,ABBC,CA,AB satisfying the following two properties: (i)(i) Lines AA1,BB1AA_1,BB_1, and CC1CC_1 are parallel. (ii)(ii) Lines AA1,BB1AA_1,BB_1, and CC1CC_1 meet dd respectively at points A′,B′A' ,B', and C′C' such that A′A1A′A=B′B1B′B=C′C1C′C\frac{A'A_1}{A' A}=\frac{B'B_1}{B 'B}=\frac{C'C_1}{C'C}
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