Perpendicular in Right-Angled Triangle makes two Similar Triangles
Theorem
In the words of Euclid:
- If in a right-angled triangle a perpendicular be drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another.
(The Elements: Book $\text{VI}$: Proposition $8$)
Proof
Let $\triangle ABC$ be a right-angled triangle such that $\angle BAC$ is a right angle.
Let $AD$ be drawn perpendicular to $BC$.
We need to show that $\triangle ABD$ is similar to $\triangle CAD$ which is similar to $\triangle CBA$ which is in turn similar to $\triangle ABD$.
We have that $\angle BAC = \angle ADB$ as both are right angles.
Then $\angle ABC$ is common to both $\triangle ABC$ and $\triangle ABD$.
So from Sum of Angles of Triangle Equals Two Right Angles:
- $\angle ACB = \angle BAD$
So $\triangle ABC$ is equiangular with $\triangle ABD$.
From Equiangular Triangles are Similar it follows that $\triangle ABC$ is similar to $\triangle ABD$.
Similarly we see that $\triangle ABC$ is equiangular with $\triangle ACD$.
Therefore each of $\triangle ABD$ and $\triangle ACD$ are similar to $\triangle ABC$.
Finally, $\triangle ABD$ and $\triangle ACD$ are equiangular with each other and so similar to each other.
$\blacksquare$
Porism
In the words of Euclid:
- From this it is clear that, if in a right-angled triangle a perpendicular be drawn from the right angle to the base, the straight line so drawn is a mean proportional between the segments of the base.
(The Elements: Book $\text{VI}$: Proposition $8$ : Porism)
Historical Note
This proof is Proposition $8$ of Book $\text{VI}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VI}$. Propositions