Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/231

Rh therefore also the parallelogram AC has to the parallelogram CF the ratio which is compounded of the ratios of the sides.

Wherefore, parallelograms &c.

PROPOSITION 24. THEOREM.

Parallelograms about the diameter of any parallelogram are similar to the whole parallelogram and to one another.

Let ABCD be a parallelogram, of which AC is a diameter; and let EG and HK be parallelograms about the diameter: the parallelograms EG and HK shall be similar both to the whole parallelogram and to one another.

For, because DC and GF are parallels, the angle ADC is equal to the angle AGF. [I. 29. And because BC and EF are parallels, the angle ABC is equal to the angle AEF. [I. 29. And each of the angles BCD and EFG is equal to the opposite angle BAD, [I. 34. and therefore they are equal to one another. Therefore the parallelograms ABCD and AEFG are equiangular to one another.

And because the angle ABC is equal to the angle AEF, and the angle BAC is common to the two triangles BAC and EAF, therefore these triangles are equiangular to one another; and therefore AB is to BC as AE is to EF. [VI. 4. And the opposite sides of parallelograms are equal to one another; therefore AB is to AD as AE is to AG, and DC is to CB as GF is to FE, and CD is to DA as FG is to GA.[V.7