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vector equality, Lemoine point

Source: 2023 Viet Nam math olympiad for high school students D1 P5

March 25, 2023
geometry

Problem Statement

Given a triangle ABCABC with Lemoine point L.L. Let a=BC,b=CA,c=AB.a=BC, b=CA,c=AB.
Prove that: a2LA→+b2LB→+c2LC→=0→.{a^2}\overrightarrow {LA} + {b^2}\overrightarrow {LB} + {c^2}\overrightarrow {LC} = \overrightarrow 0 .
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