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only 1 Fermat-Torricelli point inside a triangle

Source: 2022 Viet Nam math olympiad for high school students D1 P1

March 20, 2023
geometry

Problem Statement

Let ABCABC be a triangle with ∠A,∠B,∠C<120∘\angle A,\angle B,\angle C <120^{\circ}.
Prove that: there is exactly one point TT inside △ABC\triangle ABC such that ∠BTC=∠CTA=∠ATB=120∘\angle BTC=\angle CTA=\angle ATB=120^{\circ}. (TT is called Fermat-Torricelli point of △ABC\triangle ABC)
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