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The Orthocentre

In ABC, if ...

Question

In $△ A B C$ , if $^{'} O^{'}$ is the cicumfcentre and $H$ is the orthocentre, then show that
$H A + H B + H C = 2 H O$

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Solution

Let $D$ be the mid point of $B C$ .
Take $O$ as the origin.
We have $\to A H = 2 \to R cos A, \to O D = R cos A$
where $R$ is the circumradius of $△ A B C$ and hence $\to A H = 2 \to O D$
Now, $\to H A + \to H B + \to H C = \to H A + 2 \to H D$
$= \to H A + 2 (\to H O + \to O D)$
$= \to H A + 2 \to H O + 2 \to O D$
$= \to H A + 2 \to H O + \to A H$
$= 2 \to H O$

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