Euclid's second theorem

Euclid’s second theorem concerns the altitude relative to the hypotenuse, which divides the latter into two segments (the projections of the legs).

Theorem — Euclid's second theorem

In a right-angled triangle, the altitude relative to the hypotenuse is the geometric mean between the projections of the legs onto the hypotenuse: $\frac{AH}{CH} = \frac{CH}{BH}$

From this proportion it follows that $CH^2 = AH \cdot BH$: the square of the altitude equals the product of the two projections.

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