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vector and lengths (in)equality

Source: 2022 Viet Nam math olympiad for high school students D1 P4

March 20, 2023
geometryvector

Problem Statement

Assume that △ABC\triangle ABC is acute. Let a=BC,b=CA,c=ABa=BC, b=CA, c=AB. a) Denote HH by the orthocenter of △ABC\triangle ABC. Prove that:a.HA→HA+b.HB→HB+c.HC→HC=0→.a.\frac{{\overrightarrow {HA} }}{{HA}} + b.\frac{{\overrightarrow {HB} }}{{HB}} + c.\frac{{\overrightarrow {HC} }}{{HC}} = \overrightarrow 0 . b) Consider a point PP lying on the plane. Prove that the sum:aPa+bPB+cPCaPa+bPB+cPC get its minimum value iff P≡HP\equiv H.
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