Sides of Equiangular Triangles are Reciprocally Proportional

Theorem

As Euclid defined it:

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 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 a 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 is Proposition 15 of Book VI of Euclid's The Elements.