Question

If $A(\cos \alpha ,\sin \alpha ),$ $B(\cos \beta ,\sin \beta ),$ $C(\cos \gamma ,\sin \gamma )$ are the vertices of $△ABC$ and $H,G,S$ are the orthocentre, centroid and circumcentre of $△ABC$ respectively, then which of the following is/are true?

• A. $S\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$
• B. $G\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$
• C. $H\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$
• D. $S\equiv (0,0)$

Answer 1

Answer: (C), (D)

$S\equiv (0,0)$


$OA=OB=OC=1$ ($O$ is origin)
$\therefore$ Circumcentre, $S\equiv (0,0)$

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


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

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

$\Rightarrow H\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$