Euclid's first theorem

By combining Pythagoras’ theorem with similarity we obtain the theorems of Euclid, which relate the sides of a right-angled triangle to the segments created by the altitude on the hypotenuse.

Theorem — Euclid's first theorem

In a right-angled triangle, each leg is the geometric mean between the hypotenuse and its projection onto the hypotenuse.

In other words, letting $AB$ be the hypotenuse and $AH$ the projection of the leg $AC$ onto it, the proportion $AB : AC = AC : AH$ holds, that is $AC^2 = AB \cdot AH$.

Links

Topics: Similarity
Concepts: Geometric mean · Euclid’s theorems · Right-angled triangle
Skills: Synthetic geometry
People: Euclid · Pythagoras