Definition:Side of Sum of Medial Areas

Definition

Let $a, b \in \R_{>0}$ be in the forms:

$a = \dfrac {\rho \lambda^{1/4} } {\sqrt 2} \sqrt {1 + \dfrac k {\sqrt {1 + k^2} } }$

$b = \dfrac {\rho \lambda^{1/4} } {\sqrt 2} \sqrt {1 - \dfrac k {\sqrt {1 + k^2} } }$

where:

$\rho$ is a rational number

$k$ is a rational number whose square root is irrational

$\lambda$ is a rational number whose square root is irrational.

Then $a + b$ is the side of the sum of (two) medial areas.

In the words of Euclid:

If two straight lines incommensurable in square which make the sum of the squares on them medial, and the rectangle contained by them medial and also incommensurable with the sum of the squares on them, be added together, the whole straight line is irrational; and let it be called the side of the sum of two medial areas.

(The Elements: Book $\text{X}$: Proposition $41$)

Also see

• Construction of Components of Side of Sum of Medial Areas

• Side of Sum of Medial Areas is Irrational