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|PA_1|^2 +... + |PA_m|^2 =|PB_1|^2+...+|PA_n|^2, point P, m+n points

Source: 2015 Baltic Way Shortlist G5 BW https://artofproblemsolving.com/community/c2641329_baltic_way_shortlist__geometry

November 22, 2021
geometrypoints

Problem Statement

For what positive numbers mm and nn do there exist points A1,...,AmA_1, ..., Am and B1...,BnB_1 ..., B_n in the plane such that, for any point PP, the equation ∣PA1∣2+...+∣PAm∣2=∣PB1∣2+...+∣PAn∣2|PA_1|^2 +... + |PA_m|^2 =|PB_1|^2+...+|PA_n|^2 holds true?
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