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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
$O A + O B + O C = O H$

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Solution

Let $D$ be the mid point of $B C$ .
Take $O$ as the origin.
Let $\to O A = \to a$ , $\to O B = \to b$ and $\to O C = \to c$
From the figure. $\to O D = \frac{\to b + \to c}{2}$
$∴ \to O A + \to O B + \to O C = \to O A + 2 \to O D = \to O A + \to A H = \to O H$
Hence proved.

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