Parallelograms About Diameter are Similar

Theorem

In the words of Euclid:

In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another.

(The Elements: Book $\text{VI}$: Proposition $24$)

Proof

Let $\Box ABCD$ be a parallelogram and $AC$ one of its diameters.

Let $EG, HK$ be parallelograms about $AC$.

We have that $EF \parallel CD$.

So from Parallel Transversal Theorem:

$BE : EA = CF : FA$

Again, we have that:

$FG \parallel CD$

So from Parallel Transversal Theorem:

$CF : FA = DG : GA$

From Equality of Ratios is Transitive:

$BE : EA = DG : GA$

So from Magnitudes Proportional Separated are Proportional Compounded:

$BA : AE = DA : AG$

From Proportional Magnitudes are Proportional Alternately:

$BA : AD = EA : AG$

Therefore in the parallelograms $\Box ABCD$ and $\Box EG$, the sides about the common angle $\angle BAD$ are proportional.

We have that $GF \parallel DC$, $\angle AFG = \angle DCA$ and $\angle DAC$ is common to $\triangle ADC$ and $\triangle AGF$.

Therefore $\triangle ADC$ is equiangular with $\triangle AGF$.

For the same reason, $\triangle ACB$ is equiangular with $\triangle AFE$.

Thus the whole parallelogram $\Box ABCD$ is equiangular with the parallelogram $\Box EG$.

Therefore:

$AD : DC = AG : GF$

$DC : CA = GF : FA$

$AC : CB = AF : FE$

$CB : BA = FE : FA$

Since we also have:

$DC : CA = GF : FA$

$AC : CB = AF : FE$

it follows from Equality of Ratios Ex Aequali that:

$DC : CB = GF: FE$

Therefore in the parallelograms $\Box ABCD$ and $\Box EG$, sides about the equal angles are proportional.

Therefore from Book $\text{VI}$ Definition $1$: Similar Rectilineal Figures, $\Box ABCD$ is similar to $\Box EG$.

By the same argument, $\Box ABCD$ is similar to $\Box KH$.

Therefore by Similarity of Polygons is Equivalence Relation‎, $\Box EG$ is similar to $\Box HK$.

$\blacksquare$

Historical Note

This proof is Proposition $24$ of Book $\text{VI}$ of Euclid's The Elements.
It is the converse of Proposition $26$: Parallelogram Similar and in Same Angle has Same Diameter.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VI}$. Propositions