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]].
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All acute triangle angles are less then 90 degrees. For example, an [[equilateral triangle]] is always acute, since all angles (which are 60) are all less than 90.
  
 
Acute triangles can also be defined in different ways:  
 
Acute triangles can also be defined in different ways:  
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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]].
 
* 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]].
  
An [[equilateral triangle]] is always acute. In an acute triangle, the [[circumcenter]], [[incenter]], [[orthocenter]], and [[centroid]] are all within the interior of the triangle.  
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In an acute triangle, the [[circumcenter]], [[incenter]], [[orthocenter]], and [[centroid]] are all within the interior of the triangle.  
  
 
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Revision as of 23:07, 24 January 2016

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.

All acute triangle angles are less then 90 degrees. For example, an equilateral triangle is always acute, since all angles (which are 60) are all less than 90.

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.

In an acute triangle, the circumcenter, incenter, orthocenter, and centroid are all within the interior of the triangle.

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