Construction of Components of Major/Lemma

Proof

 * Euclid-X-33-Lemma.png

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

Let the perpendicular $AD$ be drawn.

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

From :
 * $CB : BA = BA : BD$

Therefore from :
 * the recangle contained by $CB$ and $BD$ equals the square on $AB$.

For the same reason:
 * the recangle contained by $BC$ and $CD$ equals the square on $AC$.

From :
 * $BD : DA = AD : DC$

Therefore from :
 * the recangle contained by $BD$ and $DC$ equals the square on $AD$.

From :
 * $CB : CA = BA : AD$

From :
 * the recangle contained by $BC$ and $AD$ equals the recangle contained by $BA$ and $AC$.