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:  
  
 
* 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.
* 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]]. It is a consequence of the Law of Cosines as the cosine of an angle that is less than 90 degrees is positive.
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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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Latest revision as of 18:22, 25 January 2020

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. It is a consequence of the Law of Cosines as the cosine of an angle that is less than 90 degrees is positive.

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

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