View source for Orthocenter

The '''orthocenter''' of a [[triangle]] is the point of intersection of its [[altitude]]s.  It is [[mathematical convention | conventionally]] denoted <math>H</math>.


[[Image:Orthoproof1.PNG|center]]


== Proof of Existence ==

''Note: The orthocenter's existence is a trivial consequence of the [[trigonometric version of Ceva's Theorem]]; however, the following proof, due to [[Leonhard Euler]], is much more clever, illuminating and insightful.''

Consider a triangle <math>ABC</math> with [[circumcenter]] <math>O</math> and [[centroid]] <math>G</math>.  Let <math>A'</math> be the midpoint of <math>BC</math>.  Let <math>H</math> be the point such that <math>G</math> is between <math>H</math> and <math>O</math> and <math>HG = 2 HO</math>.  Then the triangles <math>AGH</math>, <math>A'GO</math> are [[similar]] by angle-side-angle similarity.  It follows that <math>AH</math> is parallel to <math>OA'</math> and is therefore perpendicular to <math>BC</math>; i.e., it is the altitude from <math>A</math>.  Similarly, <math>BH</math>, <math>CH</math>, are the altitudes from <math>B</math>, <math>{C}</math>.  Hence all the altitudes pass through <math>H</math>.  Q.E.D.

This proof also gives us the result that the orthocenter, centroid, and circumcenter are [[collinear]], in that order, and in the proportions described above.  The line containing these three points is known as the [[Euler line]] of the triangle.

==See Also==
*[[Triangle center]]
*[[Altitude]]