Question

In $△ABC$, if ${}^{\prime }{O}^{\prime }$ is the circumcentre and $H$ is the orthocentre, then show that $OA+OB+OC=OH$.

Answer 1

we know that

$HG=2GO$ where $G$ is centroid of triangle

let a point D, between B and C

$OD=(OB+OC)/2$

$OA+OB+OC=OA+2OD$

we know that G divide The point A and midpoint

opposite side in ratio 2 :1

$OG=\frac{OA+2OD}{3}$

$OA+OB+OC=30G=20G+OG$

$=HG+OG$

$OA+OB+OC=HO$