Editing Orthocenter (section)

==Relation with circles and conics==

Denote the [[circumradius]] of the triangle by {{mvar|R}}. Then<ref>[http://mathworld.wolfram.com/Orthocenter.html Weisstein, Eric W. "Orthocenter." From MathWorld--A Wolfram Web Resource.]</ref><ref>{{harvnb|Altshiller-Court|2007|loc=p. 102}}</ref>

:<math>a^2 + b^2 + c^2 + \overline{AH}^2 + \overline{BH}^2 + \overline{CH}^2 = 12R^2.</math>

In addition, denoting {{mvar|r}} as the radius of the triangle's [[incircle]], {{mvar|r{{sub|a}}, r{{sub|b}}, r{{sub|c}}}} as the radii of its [[excircle]]s, and {{mvar|R}} again as the radius of its circumcircle, the following relations hold regarding the distances of the orthocenter from the vertices:<ref>[http://forumgeom.fau.edu/FG2006volume6/FG200639.pdf Bell, Amy, "Hansen's right triangle theorem, its converse and a generalization", ''Forum Geometricorum'' 6, 2006, 335–342.]</ref>

:<math>\begin{align}
& r_a + r_b + r_c + r = \overline{AH} + \overline{BH} + \overline{CH} + 2R, \\
& r_a^2 + r_b^2 + r_c^2 + r^2 = \overline{AH}^2 + \overline{BH}^2 + \overline{CH}^2 + (2R)^2.
\end{align}</math>

If any altitude, for example, {{mvar|{{overline|AD}}}}, is extended to intersect the circumcircle at {{mvar|P}}, so that {{mvar|{{overline|AD}}}} is a chord of the circumcircle, then the foot {{mvar|D}} bisects segment {{mvar|{{overline|HP}}}}:<ref name=pballew/>
:<math>\overline{HD} = \overline{DP}.</math>

The [[directrix (conic section)|directrices]] of all [[parabola]]s that are externally tangent to one side of a triangle and tangent to the extensions of the other sides pass through the orthocenter.<ref>Weisstein, Eric W. "Kiepert Parabola." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/KiepertParabola.html</ref>

A [[circumconic]] passing through the orthocenter of a triangle is a [[rectangular hyperbola]].<ref>Weisstein, Eric W. "Jerabek Hyperbola." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/JerabekHyperbola.html</ref>