Sides of Equiangular Triangles are Reciprocally Proportional
Theorem
In the words of Euclid:
- In equal triangles which have one angle equal to one angle the sides about the equal angles are reciprocally proportional; and those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal.
(The Elements: Book $\text{VI}$: Proposition $15$)
Note: in the above, equal is to be taken to mean of equal area.
Proof
Let $\triangle ABC, \triangle ADE$ be triangles of equal area which have one angle equal to one angle, namely $\angle BAC = \angle DAE$.
We need to show that $CA : AD = EA : AB$, that is, the sides about the equal angles are reciprocally proportional.
Place them so $CA$ is in a straight line with $AD$.
From Two Angles making Two Right Angles make Straight Line $EA$ is also in a straight line with $AB$.
Join $BD$.
It follows from Ratios of Equal Magnitudes that:
- $\triangle CAB : \triangle BAD = \triangle EAD : \triangle BAD$
But from Areas of Triangles and Parallelograms Proportional to Base:
- $\triangle CAB : \triangle BAD = CA : AD$
Also from Areas of Triangles and Parallelograms Proportional to Base:
- $\triangle EAD : \triangle BAD = EA : AB$
So from Equality of Ratios is Transitive:
- $CA : AD = EA : AB$
$\Box$
Now let the sides in $\triangle ABC, \triangle ADE$ be reciprocally proportional.
That is, $CA : AD = EA : AB$.
Join $BD$.
From Areas of Triangles and Parallelograms Proportional to Base:
- $\triangle CAB : \triangle BAD = CA : AD$
Also from Areas of Triangles and Parallelograms Proportional to Base:
- $\triangle EAD : \triangle BAD = EA : AB$
It follows from Equality of Ratios is Transitive that:
- $\triangle CAB : \triangle BAD = \triangle EAD : \triangle BAD$
So from Magnitudes with Same Ratios are Equal:
- $\triangle ABC = \triangle ADE$
$\blacksquare$
Historical Note
This proof is Proposition $15$ 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