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Locus of midpoints

Source: Czech and Slovak Olympiad 1963, National Round, Problem 3

June 16, 2024
geometryLocustangent circlescommon tangent

Problem Statement

A line MNMN is given in the plane. Consider circles k1k_1, k2k_2 tangent to the line at points MM, NN, respectively, while touching each other externally. Let XX be the midpoint of the segment PQPQ, where PP, QQ are in this order tangent points of the second common external tangent of the circles k1k_1, k2k_2. Find the locus of the points XX for all pairs of circles of the specified properties.
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