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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.

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 interior of the triangle.

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Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.
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