Obtuse triangle

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.

Obtuse triangles all have one angle larger then 90. For example a triangle with angles 10,10 and 160, it would be a obtuse angle, since 160>90.

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 $a, b$ and $c$, $c > a, b$, is obtuse if and only if $a^2 + b^2 < c^2$. This is known as the Pythagorean Inequality. It follows directly from the Law of Cosines.

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