The First Six Books of the Elements of Euclid

by Euclid; John Casey

2. Two quadrilaterals whose diagonals intersect at equal angles are to one another in the ratio of the rectangles of the diagonals.

PROP. XXIV.—Theorem.
In any parallelogram (AC), every two parallelograms (AF, FC) which are about a diagonal are similar to the whole and to one another.

Dem.—Since the parallelograms AC, AF have a common angle, they are equiangular [I. xxxiv.], and all that is required to be proved is, that the sides about the equal angles are proportional. Now, since the lines EF, BC are parallel, the triangles AEF, ABC are equiangular; therefore [iv.] AE : EF :: AB : BC, and the other sides of the parallelograms are equal to AE, EF; AB, BC: hence the sides about the equal angles are proportional; therefore the parallelograms AF, AC are similar. In the same manner the parallelograms AF, FC are similar.

Cor.—The parallelograms AF, FC, AC are, two by two, homothetic.

PROP. XXV.—Problem.
To describe a rectilineal figure equal to a given one (A), and similar to another given one (BCD).

Sol.—On any side BC of the figure BCD describe the rectangle BE equal to BCD [I. xlv.], and on CE describe the rectangle EF equal to A. Between BC, CF find a mean proportional GH, and on it describe the figure GHI similar to BCD [xviii.], so that BC and GH may be homologous sides. GHI is the figure required.

Dem.—The three lines BC, GH, CF are in continued proportion; therefore BC : CF in the duplicate ratio of BC : GH [V. Def. x.]; and since the figures BCD, GHI are similar, BCD : GHI in the duplicate ratio of BC : GH [xx.]; also BC : CF :: rectangle BE : rectangle EF. Hence rectangle BE : EF :: figure BCD : GHI; but the rectangle BE is equal to the figure BCD; therefore the rectangle EF is equal to the figure GHI; but EF is equal to A (const.). Therefore the figure GHI is equal to A, and it is similar to BCD. Hence it is the figure required.