Geometric mean theorem

area of grey square = area of grey rectangle: $h^{2}=pq\Leftrightarrow h={\sqrt {pq}}$

In Euclidean geometry, the geometric mean theorem or right triangle altitude theorem is a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of the two segments equals the altitude.

Expressed as a mathematical formula, if $h$ denotes the altitude in a right triangle and $p$ and $q$ denote the segments that the altitude creates on the hypotenuse, it can be stated as:^[1]

h={\sqrt {pq}}

or in term of areas:

h^{2}=pq

.

The converse statement is true as well. Any triangle, in which the altitude equals the geometric mean of the two line segments created by it, is a right triangle.

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