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locus related to distances from vertices of regular hexagon

Source: Switzerland - 2009 Swiss MO Final Round p1

December 26, 2022
geometryLocusgeometric inequalityhexagon

Problem Statement

Let PP be a regular hexagon. For a point AA, let d1≤d2≤...≤d6d_1\le d_2\le ...\le d_6 the distances from AA to the six vertices of PP, ordered by magnitude. Find the locus of all points AA in the interior or on the boundary of PP such that: (a) d3d_3 takes the smallest possible value. (b) d4d_4 takes the smallest possible value.
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