Editing Orthocenter (section)

==Relation to other centers, the nine-point circle==
{{main|Nine-point circle}}
The orthocenter {{mvar|H}}, the [[centroid]] {{mvar|G}}, the [[circumcenter]] {{mvar|O}}, and the center {{mvar|N}} of the [[nine-point circle]] all lie on a single line, known as the [[Euler line]].<ref>{{harvnb|Berele|Goldman|2001|loc=p. 123}}</ref> The center of the nine-point circle lies at the [[midpoint]] of the Euler line, between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half of that between the centroid and the orthocenter:<ref>{{harvnb|Berele|Goldman|2001|loc=pp. 124-126}}</ref>

:<math>\begin{align}
& \overline{OH} = 2\overline{NH}, \\
& 2\overline{OG} = \overline{GH}.
\end{align}</math>

The orthocenter is closer to the [[incenter]] {{mvar|I}} than it is to the centroid, and the orthocenter is farther than the incenter is from the centroid:

:<math>\begin{align}
\overline{HI} &< \overline{HG}, \\
\overline{HG} &> \overline{IG}.
\end{align} </math>

In terms of the sides {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, [[inradius]] {{mvar|r}} and [[circumradius]] {{mvar|R}},<ref>Marie-Nicole Gras, "Distances between the circumcenter of the extouch triangle and the classical centers", ''Forum Geometricorum'' 14 (2014), 51-61.  http://forumgeom.fau.edu/FG2014volume14/FG201405index.html</ref><ref name=SL>Smith, Geoff, and Leversha, Gerry, "Euler and triangle geometry", ''Mathematical Gazette'' 91, November 2007, 436–452.</ref>{{rp|p. 449}}

:<math>\begin{align}
\overline{OH}^2 &= R^2 -8R^2 \cos A \cos B \cos C \\
&= 9R^2-(a^2+b^2+c^2), \\
\overline{HI}^2 &= 2r^2 -4R^2 \cos A \cos B \cos C.
\end{align}</math>