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Geometric inequality in Taiwan TST

Source: 2022 Taiwan TST Round 1 Independent Study 1-G

October 26, 2022
Taiwangeometric inequalitygeometryinequalities

Problem Statement

Two triangles ABCABC and A′B′C′A'B'C' are on the plane. It is known that each side length of triangle ABCABC is not less than aa, and each side length of triangle A′B′C′A'B'C' is not less than a′a'. Prove that we can always choose two points in the two triangles respectively such that the distance between them is not less than a2+a′23\sqrt{\dfrac{a^2+a'^2}{3}}.
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