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orthocenter (Definition)

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

\includegraphics{ortho}

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.



"orthocenter" is owned by Mathprof. [ full author list (2) | owner history (2) ]
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See Also: height, median, triangle, Euler line, orthic triangle, Ceva's theorem, Ceva's theorem, triangle center, incenter, trigonometric version of Ceva's theorem, centroid

Also defines:  orthocentric tetrad
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Cross-references: cyclic, quadrilaterals, Euler line, lies on, orthic triangle, triangle centers, obtuse, right, acute, vertex, heights, intersection, point, triangle
There are 9 references to this entry.

This is version 6 of orthocenter, born on 2001-10-31, modified 2007-06-05.
Object id is 646, canonical name is Orthocenter.
Accessed 17496 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
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