# Construction of Components of Major/Lemma

## Lemma to Construction of Components of Major

In the words of Euclid:

*Let $ABC$ be a right-angled triangle having the angle $A$ right, and let the perpendicular $AD$ be drawn;*

I say that the rectangle $CB, BD$ is equal to the square on $BA$, the rectangle $BC, CD$ equal to the square on $CA$, the rectangle $BD, DC$ equal to the square on $AD$, and, further, the rectangle $BC, AD$ equal to the rectangle $BA, AC$.

(*The Elements*: Book $\text{X}$: Proposition $33$ : Lemma)

## Proof

Let $\triangle ABC$ be a right-angled triangle where $\angle A$ is right.

Let the perpendicular $AD$ be drawn.

- triangles $\triangle ABC$, $\triangle ABD$ and $\triangle ADC$ are all similar to each other.

From Proposition $4$ of Book $\text{VI} $: Equiangular Triangles are Similar:

- $CB : BA = BA : BD$

Therefore from Proposition $17$ of Book $\text{VI} $: Rectangles Contained by Three Proportional Straight Lines:

$\Box$

For the same reason:

$\Box$

- $BD : DA = AD : DC$

Therefore from Proposition $17$ of Book $\text{VI} $: Rectangles Contained by Three Proportional Straight Lines:

$\Box$

From Proposition $4$ of Book $\text{VI} $: Equiangular Triangles are Similar:

- $CB : CA = BA : AD$

From Proposition $16$ of Book $\text{VI} $: Rectangles Contained by Proportional Straight Lines:

$\blacksquare$

## Historical Note

This proof is Proposition $33$ 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