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number of points C on a line such that ABC is isosceles for fixed A,B

Source: Netherlands - Dutch NMO 1965 p4

January 31, 2023
geometryisoscelescombinatorial geometry

Problem Statement

Given are the points AA and BB in the plane. If xx is a straight line is in that plane, and xx does not coincide with the perpendicular bisectror of ABAB, then denote the number of points CC located at xx such that △ABC\vartriangle ABC is isosceles, as the "weight of the line xx”. Prove that the weight of any line xx is at most 55 and determine the set of points PP which has a line with weight 11, but none with weight 00.
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