Construction of Figure Similar to One and Equal to Another

Theorem

As Euclid defined it:

To construct one and the same figure similar to a given rectilineal figure and equal to another given rectilineal figure.

(The Elements: Book VI: Proposition $25$)

Construction

Let $ABC$ be the given rectilineal figure to which the figure to be constructed is to be similar, and let $D$ be the rectilineal figure to which it must be equal (in area).

Using Construction of Parallelogram on Given Line Equal to Triangle in Given Angle, we apply the parallelogram $BE$ to the straight line $BC$ equal in area to $\triangle ABC$.

Using Construction of Parallelogram in Given Angle Equal to Given Polygon we apply the parallelogram $CM$ to the straight line $CE$ equal in area to $D$, where $\angle FCE = \angle CBL$.

Now using Construction of Mean Proportional‎ we construct the straight line $GH$ such that $BC : GH = GH : CF$.

Using Construction of Similar Polygon we construct $\triangle KGH$ to be similar to $\triangle ABC$ and similarly situated.

Then $\triangle KGH$ is the required rectilineal figure, similar to $\triangle ABC$ and equal in size to $D$.

Proof

We have that $BC$ is in a straight line with $CF$ and that $LE$ is in a straight line with $EM$.

From the porism to Ratio of Areas of Similar Triangles, $BC : CF = \triangle ABC : \triangle KGH$.

But from Areas of Triangles and Parallelograms Proportional to Base, $BC : CF = \Box BE : \Box EF$.

So $\triangle ABC : \triangle KGH = \Box BE : \Box EF$.

So from Proportional Magnitudes are Proportional Alternately $\triangle ABC : \Box BE = \triangle KGH : \Box EF$.

But $\triangle ABC = \Box BE$ and so $\triangle KGH = \Box EF$.

But $\Box EF = D$, and so $\triangle KGH = D$, and $\triangle KGH$ is similar to $\triangle ABC$.

Hence the result.

$\blacksquare$

Historical Note

This is Proposition 25 of Book VI of Euclid's The Elements.