Difference between revisions of "Acute triangle"

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An '''acute triangle''' is a [[triangle]] in which each [[angle]] is an [[acute angle]].  Any triangle which is not acute is either a [[right triangle]] or an [[obtuse triangle]].
 
An '''acute triangle''' is a [[triangle]] in which each [[angle]] is an [[acute angle]].  Any triangle which is not acute is either a [[right triangle]] or an [[obtuse triangle]].
 
  
 
Acute triangles can also be defined in different ways:  
 
Acute triangles can also be defined in different ways:  
  
 
* A triangle is acute if and only if each of its [[altitude]]s lies entirely in the triangle's interior.
 
* A triangle is acute if and only if each of its [[altitude]]s lies entirely in the triangle's interior.
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* A triangle with sides of length <math>a, b</math> and <math>c</math> is acute if and only if <math>a^2 + b^2 > c^2</math>, <math>b^2 + c^2 > a^2</math> and <math>c^2 + a^2 > b^2</math>. This is known as the [[Geometric inequality | Pythagorean Inequality]].
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* A triangle with sides of length <math>a, b</math> and <math>c</math> is acute if and only if <math>a^2 + b^2 > c^2</math>, <math>b^2 + c^2 > a^2</math> and <math>c^2 + a^2 > b^2</math>.  This is known as the [[Geometric inequality | Pythagorean Inequality]].
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[[Category:Definition]]
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[[Category:Geometry]]

Revision as of 22:53, 20 April 2008

An acute triangle is a triangle in which each angle is an acute angle. Any triangle which is not acute is either a right triangle or an obtuse triangle.

Acute triangles can also be defined in different ways:

  • A triangle is acute if and only if each of its altitudes lies entirely in the triangle's interior.
  • A triangle with sides of length $a, b$ and $c$ is acute if and only if $a^2 + b^2 > c^2$, $b^2 + c^2 > a^2$ and $c^2 + a^2 > b^2$. This is known as the Pythagorean Inequality.

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