Difference between revisions of "Orthocenter"

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The '''orthocenter''' of a [[triangle]] is the point of intersection of its [[altitude]]s.  It is [[mathematical convention | conventionally]] denoted <math>\displaystyle H</math>.
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The '''orthocenter''' of a [[triangle]] is the point of intersection of its [[altitude]]s.  It is [[mathematical convention | conventionally]] denoted <math>H</math>.
  
  
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''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.''
 
''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>\displaystyle ABC</math> with [[circumcenter]] <math>\displaystyle O</math> and [[centroid]] <math>\displaystyle G</math>.  Let <math>\displaystyle A'</math> be the midpoint of <math>\displaystyle BC</math>.  Let <math>\displaystyle H</math> be the point such that <math>\displaystyle G</math> is between <math>\displaystyle H</math> and <math>\displaystyle O</math> and <math>\displaystyle HG = 2 HO</math>.  Then the triangles <math>\displaystyle AGH</math>, <math>\displaystyle A'GO</math> are [[similar]] by angle-side-angle similarity.  It follows that <math>\displaystyle AH</math> is parallel to <math>\displaystyle OA'</math> and is therefore perpendicular to <math>\displaystyle BC</math>; i.e., it is the altitude from <math>\displaystyle A</math>.  Similarly, <math>\displaystyle BH</math>, <math>\displaystyle CH</math>, are the altitudes from <math>\displaystyle B</math>, <math>\displaystyle {C}</math>.  Hence all the altitudes pass through <math>\displaystyle H</math>.  Q.E.D.
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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.
 
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.
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==See Also==
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*[[Triangle center]]
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*[[Altitude]]

Revision as of 19:57, 24 November 2007

The orthocenter of a triangle is the point of intersection of its altitudes. It is conventionally denoted $H$.


Orthoproof1.PNG


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