Difference between revisions of "Acute triangle"

Revision as of 13:25, 9 February 2009

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.

An equilateral triangle is always acute. In an acute triangle, the circumcenter, incenter, orthocenter, and centroid are all within the triangle.

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 * 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]].
+An [[equilateral triangle]] is always acute. In an acute triangle, the [[circumcenter]], [[incenter]], [[orthocenter]], and [[centroid]] are all within the triangle.
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