Difference between revisions of "Obtuse triangle"
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* A triangle is obtuse if and only if two of its [[altitude]]s lie entirely outside the triangle. (There is no triangle with exactly one altitude or all three altitudes outside the triangle.) | * A triangle is obtuse if and only if two of its [[altitude]]s lie entirely outside the triangle. (There is no triangle with exactly one altitude or all three altitudes outside the triangle.) | ||
| − | * A triangle with sides of length <math>a, b</math> and <math>c</math>, <math>c > a, b</math>, is obtuse if and only if <math>a^2 + b^2 < c^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>, <math>c > a, b</math>, is obtuse if and only if <math>a^2 + b^2 < c^2</math>. This is known as the [[Geometric inequality | Pythagorean Inequality]]. It follows directly from the [[Law of Cosines]]. |
Revision as of 12:25, 8 December 2006
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An obtuse triangle is a triangle in which one angle is an obtuse angle. Any triangle which is not obtuse is either a right triangle or an acute triangle.
The obtuse triangles can also be defined in different ways:
- A triangle is obtuse if and only if two of its altitudes lie entirely outside the triangle. (There is no triangle with exactly one altitude or all three altitudes outside the triangle.)
- A triangle with sides of length
and 
, 
, is obtuse if and only if 
. This is known as the Pythagorean Inequality. It follows directly from the Law of Cosines.
and 
, 
, is obtuse if and only if 
. This is known as the 
