Inequalities for triangle elements.

Inequalities for triangle elements describe how the sides, angles, heights, medians, and other parts of a triangle are related by size. These inequalities set boundaries on how large or small these elements can be compared to each other.

Important properties

• Triangle Inequality: For any triangle with sides $a$, $b$, and $c$, $a + b > c$, $a + c > b$, and $b + c > a$.

• The side opposite a larger angle is longer than the side opposite a smaller angle.

• The length of a median or altitude is always less than the sum of the two sides it connects.

• The area of a triangle is always less than $(1/2)ab$ for sides $a$ and $b$.

• The inradius $r$ and circumradius $R$ satisfy $r < (a + b + c)/6$ and $R > a/2$ for the largest side $a$.