Construction of Components of Major

Proof

 * Euclid-X-33.png

From :

Let $AB$ and $BC$ be rational straight lines which are commensurable in square only such that:
 * $AB^2 = BC^2 + \rho^2$

such that $\rho$ is incommensurable in length with $AB$.

Let $BC$ be bisected at $D$.

From Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram:

Let a parallelogram be applied to $AB$ equal to the square on either of $BD$ or $DC$, and deficient by a square.

Let this parallelogram be the rectangle contained by $AE$ and $EB$.

Let the semicircle $AFB$ be drawn with $AB$ as the diameter.

Let $EF$ be drawn perpendicular to $AB$.

Join $AF$ and $FB$.

We have that $AB > BC$ such that $AB^2 = BC^2 + \rho^2$ such that $\rho$ is incommensurable in length with $AB$.

We also have that the rectangle contained by $AE$ and $EB$ equals the parallelogram on $AB$ equal to $\dfrac {BC} 4$.

Thus from Condition for Incommensurability of Roots of Quadratic Equation:
 * $AE$ is incommensurable in length with $EB$.

We have that:
 * $AE : BE = AB \cdot AE : AB \cdot BE$

From :
 * $AB \cdot AE = AF^2$

and:
 * $AB \cdot BE = BF^2$

Therefore $AF^2$ and $BF^2$ are incommensurable.

By definition, $AF$ and $BF$ are therefore incommensurable in square.

As $AB$ is a rational straight line, it follows by definition that $AB^2$ is a rational area.

From Pythagoras's Theorem:
 * $AB^2 = \left({AF + FB}\right)^2$

Thus $\left({AF + FB}\right)^2$ is also a rational area.

Therefore $AF + FB$ is rational.

From :
 * $AE \cdot EB = EF^2$

and by hypothesis:
 * $AE \cdot EB = BD^2$

then:
 * $FE = BD$

Therefore:
 * $BC = 2 FE$

Thus $AB \cdot BC$ is commensurable with $AB \cdot EF$.

But from Medial is Irrational:
 * $AB \cdot BC$ is medial.

Therefore by :
 * $AB \cdot EF$ is also medial.

But from :
 * $AB \cdot EF = AF \cdot FB$

Therefore $AF \cdot FB$ is also medial.

But it has been proved that $AF + FB$ is rational.

Therefore we have found two rational straight lines which are incommensurable in square whose sum of squares is rational, but such that the rectangle contained by them is medial.

Also see

 * Definition:Major (Euclidean)