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Point A outside a circle (O) with tangents AB,AC

Source: Vietnamese TST 2011 P2

April 27, 2011
geometrysimilar trianglesgeometry proposed

Problem Statement

AA is a point lying outside a circle (O)(O). The tangents from AA drawn to (O)(O) meet the circle at B,C.B,C. Let P,QP,Q be points on the rays AB,ACAB, AC respectively such that PQPQ is tangent to (O).(O). The parallel lines drawn through P,QP,Q parallel to CA,BA,CA, BA, respectively meet BCBC at E,F,E,F, respectively. (a)(a) Show that the straight lines EQEQ always pass through a fixed point M,M, and FPFP always pass through a fixed point N.N. (b)(b) Show that PMā‹…QNPM\cdot QN is constant.
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