# 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 $BA, AC$.

## 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.
$CB : BA = BA : BD$
the recangle contained by $CB$ and $BD$ equals the square on $AB$.

$\Box$

For the same reason:

the recangle contained by $BC$ and $CD$ equals the square on $AC$.

$\Box$

$BD : DA = AD : DC$
the recangle contained by $BD$ and $DC$ equals the square on $AD$.

$\Box$

$CB : CA = BA : AD$
the recangle contained by $BC$ and $AD$ equals the recangle contained by $BA$ and $AC$.

$\blacksquare$

## Historical Note

This proof is Proposition $33$ of Book $\text{X}$ of Euclid's The Elements.