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