The orthocenterMathworldPlanetmath of a triangleMathworldPlanetmath is the point of intersectionMathworldPlanetmath 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 centersMathworldPlanetmath and it is very related with the orthic triangleMathworldPlanetmath (formed by the three height’s foots). It lies on the Euler lineMathworldPlanetmath and the four quadrilateralsMathworldPlanetmath 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 IncenterMathworldPlanetmath
Related topic TrigonometricVersionOfCevasTheorem
Related topic Centroid
Defines orthocentric tetrad