Question

If O is the circumcentre and O' is the orthocentre of a triangle ABC and if $\overline{AP}$ is the circumdiameter then $\overline{A{O}^{\prime }}+\overline{O^{\prime }B}+\overline{O^{\prime }C}=$ ?

• A. $\overline{OA}$
• B. $\overline{O^{\prime }A}$
• C. $\overline{AP}$
• D. $\overline{AO}$

Answer 1

The correct option is C $\overline{AP}$

$O$ is the circumcentre and $O^{\prime }$ is the orthocentre of the triangle.

Thus $2OD=A{O}^{\prime }$

Therefore, $2\overrightarrow{OD}=\overrightarrow{A{O}^{\prime }}$ ....(i)

Now take $\overrightarrow{A{O}^{\prime }}+\overrightarrow{O^{\prime }B}+\overrightarrow{O^{\prime }C}$

$=2\overrightarrow{A{O}^{\prime }}+(\overrightarrow{O^{\prime }A}+\overrightarrow{O^{\prime }B+\overrightarrow{O^{\prime }C}})$

$=2\overrightarrow{A{O}^{\prime }}+2\overrightarrow{O^{\prime }O}=2\overrightarrow{AO}$

$=\overrightarrow{AP}$ ....where $AP$ is the diameter through $A$ of the circumcircle.