# orthocenter

The orthocenter of a triangle is the point of intersection of its three heights.

In the figure, $H$ is the orthocenter of $ABC$.

The orthocenter $H$ lies inside, on a vertex or outside the triangle depending on the triangle being acute, right or obtuse respectively. Orthocenter is one of the most important triangle centers and it is very related with the orthic triangle (formed by the three height’s foots). It lies on the Euler line and the four quadrilaterals $FHDB,CHEC,AFHE$ are cyclic.

In fact,

• $A$ is the orthocenter of $B,C,H$;

• $B$ is the orthocenter of $A,C,H$;

• $C$ is the orthocenter of $A,B,H$.

The four points $A,B,C$, and $H$ form what is called an orthocentric tetrad.

 Title orthocenter Canonical name Orthocenter Date of creation 2013-03-22 11:55:41 Last modified on 2013-03-22 11:55:41 Owner Mathprof (13753) Last modified by Mathprof (13753) Numerical id 11 Author Mathprof (13753) Entry type Definition Classification msc 51-00 Related topic HeightOfATriangle Related topic Median Related topic Triangle Related topic EulerLine Related topic OrthicTriangle Related topic CEvasTheorem Related topic CevasTheorem Related topic CenterOfATriangle Related topic Incenter Related topic TrigonometricVersionOfCevasTheorem Related topic Centroid Defines orthocentric tetrad