Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial

Theorem

In the words of Euclid:

The rectangle contained by medial straight lines commensurable in length is medial.

(The Elements: Book $\text{X}$: Proposition $24$)

Proof

Let $x$ and $\lambda x$ be two medials such that:

$x \frown \lambda x$

where $\frown$ denotes that $x$ and $\lambda x$ are commensurable in length.

By Areas of Triangles and Parallelograms Proportional to Base:

$x^2 : x \cdot \lambda x = x : \lambda x$

From Commensurability of Elements of Proportional Magnitudes:

$x^2 \frown x \cdot \lambda x$

From Medial is Irrational we have that $x^2$ is medial.

Therefore by Straight Line Commensurable with Medial Straight Line is Medial: Porism:

$x \cdot \lambda x$ is medial.

$\blacksquare$

Historical Note

This proof is Proposition $24$ of Book $\text{X}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions