Question

If O is the circumcentre and P the orthocentre of $\Delta$ABC, then $\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}=?$

• A. $\overrightarrow{OP}$
• B. $2\overrightarrow{PO}$
• C. $3\overrightarrow{PO}$
• D. $4\overrightarrow{PO}$

Answer 1

The correct option is A $\overrightarrow{OP}$

We know that,

$PG=2G0$ where $G$ is the centroid of the triangle.

Let a point $D$ between $B$ and $C$

So, $OD=(OB+OC)/2$

now, $OA+OB+OC=OA+20D$

We know that $G$ divide the point $A$ and mid-point of appoicte side $D$ in ratio $2:1$

So, $OG=\frac{(OA+20D)}{2+1}$

So, $OA+OB+OC=30G$

$=20G+OG$

$=GP+OG=OP$