Geometric mean theorem

In a right triangle, the altitude to the hypotenuse is the geometric mean of the two hypotenuse segments it creates: its square equals their product.

Altitude-on-hypotenuse theorem: In a right triangle, the altitude to the hypotenuse is the geometric mean of the two segments it creates. This is the theorem’s standard name and its exact geometric setup.
Right triangle: A triangle containing one angle of 90 degrees. The theorem applies to this triangle and its altitude to the hypotenuse.
Geometric mean construction: A straightedge-and-compass construction that produces the geometric mean of two positive lengths. A semicircle construction reproduces the altitude relation to construct that mean.
Leg theorem: In a right triangle, each leg squared equals the hypotenuse times the adjacent hypotenuse segment. It relates a leg to a projection, whereas this theorem relates the altitude to both segments.
Similar triangles: Triangles with equal corresponding angles and proportional corresponding sides. The three triangles formed by the altitude are pairwise similar, yielding the squared-length relation.
Hypotenuse: The side opposite the right angle in a right triangle. The altitude lands on this side and divides it into the two relevant segments.
Right triangle trigonometry: The study of angle and side relationships in right triangles using trigonometric functions. The altitude theorem provides side lengths that can then determine trigonometric ratios.
Pythagorean theorem converse: A theorem stating that a triangle with side squares satisfying the Pythagorean equation is right-angled. It tests whether a triangle is right, rather than relating an altitude to hypotenuse segments.
Geometric mean: The positive square root of the product of two positive numbers. The altitude’s length is precisely the geometric mean of the two hypotenuse segments.
Altitude (triangle): A perpendicular segment from a triangle’s vertex to the line containing the opposite side. The theorem concerns the altitude drawn from the right angle to the hypotenuse.

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