Triangle inequality (other)

The triangle inequality can be extended to other geometric quantities beyond side lengths. For example, inequalities can relate the medians, altitudes, or angle bisectors of a triangle, or compare the perimeter and area. These inequalities often state that the sum of two such quantities is greater than the third, or that certain combinations are bounded above or below.

Important properties

• For any triangle, the sum of the lengths of any two medians is greater than the length of the third median.

• The sum of the lengths of any two altitudes is greater than the third altitude.

• Similar inequalities can be written for angle bisectors and other triangle elements.

• These inequalities are often proven using the classical triangle inequality or geometric constructions.