Question

If $S$ is circumcenter, $G$ the centroid, $O$ the orthocenter of $\Delta ABC$, then $SA+SB+SC$ is equal to:

• A. $SG$
• B. $OS$
• C. $SO$
• D. $OG$

Answer 1

Answer: (C)

$SO$

This can be easily solved by using Vectors

Let $S\left(0\right);A;B;C$ be the vectors of circumcenter(origin), $A,B,C$

$SA+SB+SC$

Centroid $G=\frac{SA+SB+SC}{3}$

We know that in a triangle $HG:GS=2:1$ by Euler's rule

O--------G----S(0)

So here $G$ divides $HS$ in $2:1$ ratio.

$\Rightarrow SG=\frac{SO+2S}{3}$

$\Rightarrow O=3SG=SA+SB+SC$

$\Rightarrow SA+SB+SC=SO$