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Sum of product of distances is constant

Source: Bundeswettbewerb Mathematik 2020, Round 1 - Problem 3

November 17, 2020
geometrygeometry proposeddistancesdiameter

Problem Statement

Let ABAB be the diameter of a circle kk and let EE be a point in the interior of kk. The line AEAE intersects kk a second time in C≠AC \ne A and the line BEBE intersects kk a second time in D≠BD \ne B.
Show that the value of AC⋅AE+BD⋅BEAC \cdot AE+BD\cdot BE is independent of the choice of EE.
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