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,

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$A$ is the orthocenter of $B,C,H$;

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$B$ is the orthocenter of $A,C,H$;

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$C$ 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  20130322 11:55:41 
Last modified on  20130322 11:55:41 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  11 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 5100 
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 