Triangle Inequality

The Triangle Inequality says that in a nondegenerate triangle $\displaystyle ABC$ :

$\displaystyle AB + BC > AC$

$\displaystyle BC + AC > AB$

$\displaystyle AC + AB > BC$

That is, the sum of the lengths of any two sides is larger than the length of the third side. In degenerate triangles, the strict inequality must be replaced by "greater than or equal to."

The Triangle Inequality can also be extended to other polygons. The lengths $a_1, a_2, \ldots, a_n$ can only be the sides of a nondegenerate $n$ -gon if $a_i < a_1 + \ldots + a_{i -1} + a_{i + 1} + \ldots + a_n = \left(\sum_{j=1}^n a_j\right) - a_i$ for $i = 1, 2 \ldots, n$ . Expressing the inequality in this form leads to $2a_i < P$ , where $P$ is the sum of the $a_j$ , or $a_i < \frac{P}{2}$ . Stated in another way, it says that in every polygon, each side must be smaller than the semiperimeter.

Example Problems

Introductory Problems

2006 AMC 10B Problem 10

2006 AIME II Problem 2

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Triangle Inequality

Example Problems

Introductory Problems

See Also