Question

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

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

Answer 1

Answer: (C)

$\overrightarrow{AP}$

$\left|\overrightarrow{AP}\right|=2\left|AO\right|$
$=O^{\prime }B+O^{\prime }C+O^{\prime }A-O^{\prime }A+A{O}^{\prime }$
$=(\overline{A}+\overline{B}+\overline{C}-3\overline{O^{\prime }})+2\overline{A{O}^{\prime }}$
So use $3\overline{OS}=2\overline{OC}$
where $S$ is centroid
$C$ is circumcentre
$O$ is orthocenter
$=3\left(\frac{\overline{A}+\overline{B}+\overline{C}-\overline{O^{\prime }}}{3}\right)+2\overline{A{O}^{\prime }}$
$=2\overline{O^{\prime }O}+2\overline{A{O}^{\prime }}$
$=2\overline{AO}$
$=\overline{AP}$