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I’ve already proven that if the Simson line goes through its pole then the pole is at a vertex. Similarly, I’ve proven that if it is perpendicular to one of its sides then the pole must again be at a vertex. How can I prove this for when it goes through the orthocenter? Thanks in advance

Alex D
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  • If the pole is not a vertex then Simpson line bisects $PH$, with $P$ being the pole and $H$ the orthocenter, so it can't contain $H$. – SMM May 05 '18 at 10:33

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