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Relation between O,G,C

If A cosα, si...

Question

If $A (cos α, sin α),$ $B (cos β, sin β),$ $C (cos γ, sin γ)$ are the vertices of $△ A B C$ and $H, G, S$ are the orthocentre, centroid and circumcentre of $△ A B C$ respectively, then which of the following is/are true?

S \equiv (\sum cos α, \sum sin α)

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G \equiv (\sum cos α, \sum sin α)

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H \equiv (\sum cos α, \sum sin α)

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S \equiv (0, 0)

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Solution

The correct options are
C $H \equiv (\sum cos α, \sum sin α)$
D $S \equiv (0, 0)$

$O A = O B = O C = 1$ ( $O$ is origin)
$∴$ Circumcentre, $S \equiv (0, 0)$

Let $H (l, m)$ be the orthocentre.

$∴ (\frac{l + 2 (0)}{3}, \frac{m + 2 (0)}{3}) \equiv G$

$\Rightarrow l \equiv \sum cos α, m = \sum sin α$
$(∵ G \equiv (\frac{\sum cos α}{3}, \frac{\sum sin α}{3}))$

$\Rightarrow H \equiv (\sum cos α, \sum sin α)$

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