Question

Let $H$ be the orthocentre of an acute-angled triangle $ABC$ and $O$ be its circumcenter. Then $\overline{HA}+\overline{HB}+\overline{HC}$

• A. Is equal to $\overline{HO}$
• B. Is equal to $3\overline{HO}$
• C. Is equal to $2\overline{HO}$
• D. Is not a scalar multiple of $\overline{HO}$ in general

Answer 1

The correct option is C Is equal to $2\overline{HO}$

$\underset{G}{\to }=\frac{\underset{A}{\to }+\underset{B}{\to }+\underset{C}{\to }}{3}$

Centriod divides hte line segment joining the circumcentre and othhocenter in $2:1$

$\Rightarrow \underset{G}{\to }=\frac{\underset{2O}{\to }+\underset{H}{\to }}{3}\underset{2O}{\to }+\underset{H}{\to }=\underset{3G}{\to }\underset{HA}{\to }+\underset{HB}{\to }+\underset{HC}{\to }=\underset{A}{\to }-\underset{H}{\to }+\underset{B}{\to }-\underset{H}{\to }+\underset{C}{\to }-\underset{H}{\to }\underset{HA}{\to }+\underset{HB}{\to }+\underset{HC}{\to }=\underset{A}{\to }+\underset{B}{\to }+\underset{C}{\to }-\underset{3H}{\to }\underset{HA}{\to }+\underset{HB}{\to }+\underset{HC}{\to }=\underset{3G}{\to }-\underset{3H}{\to }\underset{HA}{\to }+\underset{HB}{\to }+\underset{HC}{\to }=\underset{2O}{\to }+\underset{H}{\to }-\underset{3H}{\to }\underset{HA}{\to }+\underset{HB}{\to }+\underset{HC}{\to }=\underset{2O}{\to }-\underset{2H=}{\to }\underset{2HO}{\to }$

So option $C$ is correct