PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (236)
Orphanage
Unclass'd (1)
Unproven (541)
Corrections (47)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Copyright
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
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) ]
(view preamble | get metadata)

View style:

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
Log in to rate this entry.
(view current ratings)

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 17307 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)