Parallelograms About Diameter are Similar

Theorem

As Euclid defined it:

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

(The Elements: Book 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 Line in Triangle Cuts Sides Proportionally, $BE : EA = CF : FA$.

Again, we have that $FG \parallel CD$.

So from Parallel Line in Triangle Cuts Sides Proportionally, $CF : FA = DG : GA$.

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 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, $\Box EG$ is similar to $\Box HK$.

$\blacksquare$

Historical Note

This is Proposition 24 of Book VI of Euclid's The Elements.

This is the converse of Book VI Proposition 26: Parallelogram Similar and in Same Angle has Same Diameter.