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Cosines Relation with H, O and R - [IMO LongList 1971]

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January 1, 2011
trigonometrygeometrycircumcircle

Problem Statement

In a triangle ABCABC, let HH be its orthocenter, OO its circumcenter, and RR its circumradius. Prove that:
(a) ∣OH∣=R1−8cos⁡α⋅cos⁡β⋅cos⁡γ|OH| = R \sqrt{1-8 \cos \alpha \cdot \cos \beta \cdot \cos \gamma} where α,β,γ\alpha, \beta, \gamma are angles of the triangle ABC;ABC;
(b) O≡HO \equiv H if and only if ABCABC is equilateral.
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