PEM Level C 9 January 2010 Topics in Geometry - Power of a Point and the Radical Axis Theorems: 1. Let P be any point, and Σ be a given circle. Let ‘ be a line through P that intersects Σ at points A and B . (a) If the P is inside Σ , then P ( P, Σ) =-PA · PB . (b) If the P is outside Σ , then P ( P, Σ) = PA · PB . 2. Let Σ 1 and Σ 2 be two non-concentric circles. Then the locus of all points P such that P ( P, Σ 1 ) = P ( P, Σ 2 ) is a line. *The line is called that radical axis of the non-concentric circles Σ 1 and Σ 2 . 3. The radical axis of two circles that intersect at two distinct points is the line that passes through the points of intersection of the circles. Also, the radical axis of two externally tangent circles is the common internal tangent of the circles. 4. If the centers of three circles are non-collinear, then the radical axes of the pairwise circles are concurrent. Exercises: 1. Given three distinct points A , B , and C , prove that the locus of all points
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