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fermat point, vector (in)equalities

Source: 2022 Viet Nam math olympiad for high school students D1 P3

March 20, 2023
geometry

Problem Statement

Let ABCABC be a triangle with ∠A,∠B,∠C<120∘\angle A,\angle B,\angle C <120^{\circ}, TT is its Fermat-Torricelli point. Consider a point PP lying on the same plane with △ABC\triangle ABC. Prove that: a)TA→TA+TB→TB+TC→TC=0→.\dfrac{\overrightarrow {TA}}{TA}+\dfrac{\overrightarrow {TB}}{TB}+\dfrac{\overrightarrow {TC}}{TC}=\overrightarrow {0}. b)PA+PB+PC≥PA→.TA→TA+PB→.TB→TB+PC→.TC→TC.PA + PB + PC \ge \frac{{\overrightarrow {PA} \overrightarrow {.TA} }}{{TA}} + \frac{{\overrightarrow {PB} .\overrightarrow {TB} }}{{TB}} + \frac{{\overrightarrow {PC} \overrightarrow {.TC} }}{{TC}}. c)PA+PB+PC≥TA+TB+TCPA + PB + PC \ge TA + TB + TCand the equality occurs iff P≡TP\equiv T.
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