Geometric mean theorem

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

If h denotes the altitude in a right triangle and p and q the segments on the hypotenuse then the theorem can be stated as:[1] or in term of areas: The converse statement is true as well.

[1] The formulation in terms of areas yields a method to square a rectangle with ruler and compass, that is to construct a square of equal area to a given rectangle.

For such a rectangle with sides p and q we denote its top left vertex with D (see the Proof > Based on similarity section for a graphic of the construction).

Then we erect a perpendicular line to the diameter in D that intersects the half circle in C. Due to Thales' theorem C and the diameter form a right triangle with the line segment DC as its altitude, hence DC is the side of a square with the area of the rectangle.

[1] Another application of this theorem provides a geometrical proof of the AM–GM inequality in the case of two numbers.

360–280 BC), who stated it as a corollary to proposition 8 in book VI of his Elements.

Euclid however provides a different slightly more complicated proof for the correctness of the construction rather than relying on the geometric mean theorem.

the triangles △ADC, △BDC have an angle of equal size and have corresponding pairs of legs with the same ratio.

Since both arrangements yield the same triangle, the areas of the square and the rectangle must be identical.