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I know how to find the orthocenter, but how do do I check to see if I have the right answer? For instance, once I find the centroid of a triangle I can use the centroid formula(a+b+c)/3 to check my answer. Is there anything like that for the orthocenter?

moh abdi
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If $|a|=|b|=|c|$, the orthocenter is given by $a+b+c$.

user133281
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  • Is this the only way to check it? I assume you're talking about an equilateral triangle? – moh abdi Oct 27 '15 at 13:27
  • No, about a triangle which has the origin as its circumcenter. – user133281 Oct 27 '15 at 14:16
  • Do you have a visual example? – moh abdi Oct 27 '15 at 14:33
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    For instance, consider $a=(0,1)$, $b=(1,0)$ and $c=(\frac35,\frac45)$ (all lying on the unit circle, so $|a|=|b|=|c|=1$. Its orthocenter is given by $a+b+c=(\frac85, \frac95)$. – user133281 Oct 27 '15 at 14:35
  • Could you tell me how/why this works in the way that it does? Thanks so much. – moh abdi Oct 27 '15 at 15:01
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    You can show that this works by checking that the line through $a$ and $a+b+c$ is perpendicular to the line through $b$ and $c$ (by using inner products). Alternatively, you can use that the centroid, circumcenter and orthocenter of any triangle are collinear (Euler's line), with the centroid dividing the segment joining the orthocenter and circumcenter in the ratio $1:2$. – user133281 Oct 27 '15 at 15:38