Editing Orthocenter (section)

==Properties==

Let {{mvar|D, E, F}} denote the feet of the altitudes from {{mvar|A, B, C}} respectively. Then:

*The product of the lengths of the segments that the orthocenter divides an altitude into is the same for all three altitudes:<ref>{{harvnb|Johnson|2007|loc=p. 163, Section 255}}</ref><ref name=pballew>{{Cite web |url=http://www.pballew.net/orthocen.html |title="Orthocenter of a triangle" |access-date=2012-05-04 |archive-url=https://web.archive.org/web/20120705102919/http://www.pballew.net/orthocen.html |archive-date=2012-07-05 |url-status=usurped }}</ref>
:<math>\overline{AH} \cdot \overline{HD} = \overline{BH} \cdot \overline{HE} = \overline{CH} \cdot \overline{HF}.</math>
:The circle centered at {{mvar|H}} having radius the square root of this constant is the triangle's [[polar circle (geometry)|polar circle]].<ref>{{harvnb|Johnson|2007|loc=p. 176, Section 278}}</ref>

*The sum of the ratios on the three altitudes of the distance of the orthocenter from the base to the length of the altitude is 1:<ref name=Panapoi>[http://jwilson.coe.uga.edu/EMAT6680Fa06/Panapoi/assignment_8/assignment8.htm Panapoi, Ronnachai, "Some properties of the orthocenter of a triangle"], [[University of Georgia]].</ref> (This property and the next one are applications of a [[Cevian#Ratio properties|more general property]] of any interior point and the three [[cevian]]s through it.)
:<math>\frac{\overline{HD}}{\overline{AD}} + \frac{\overline{HE}}{\overline{BE}} + \frac{\overline{HF}}{\overline{CF}} = 1.</math>

*The sum of the ratios on the three altitudes of the distance of the orthocenter from the vertex to the length of the altitude is 2:<ref name=Panapoi/>
:<math>\frac{\overline{AH}}{\overline{AD}} + \frac{\overline{BH}}{\overline{BE}} + \frac{\overline{CH}}{\overline{CF}} = 2.</math>

*The [[isogonal conjugate]] of the orthocenter is the [[circumcenter]] of the triangle.<ref>{{harvnb|Smart|1998|loc=p. 182}}</ref>
*The [[isotomic conjugate]] of the orthocenter is the [[Lemoine point|symmedian point]] of the [[Medial triangle#Anticomplementary triangle|anticomplementary triangle]].<ref>Weisstein, Eric W. "Isotomic conjugate" From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IsotomicConjugate.html</ref>
*Four points in the plane, such that one of them is the orthocenter of the triangle formed by the other three, is called an [[orthocentric system]] or orthocentric quadrangle.