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