If ABK : CDL :: EI : GJ, AB : CD :: EF : GH.
| 3 Dem. 2.—ABK : CDL :: AB2 : CD2 [xx.], and EI : GJ :: EF2 : GH2 [xx.]; therefore AB2 : CD2 :: EF2 : GH2. Hence AB : CD :: EF : GH. |
The enunciation of this Proposition is wrongly stated in Simson’s Euclid, and in those that copy it. As given in those works, the four figures should be similar.
PROP. XXIII.—Theorem.
Equiangular parallelograms (AD, CG) are to each other as the rectangles contained by their sides about a pair of equal angles.
Dem.—Let the two sides AB, BC about the equal angles ABD, CBG, be placed so as to form one right line; then it is evident, as in Prop. xiv., that GB, BD form one right line. Complete the parallelogram BF. Now, denoting the parallelograms AB, BF, CG by X, Y , Z, respectively, we have—
| X | : Y :: AB : BC [i.], | ||||||||||
| Y | : Z :: BD : BG [i.]. | ||||||||||
| Hence | XY | : Y Z :: AB.BD : BC.BG; | |||||||||
| or | X | : Z :: AB.BD : BC.BG. |
Observation.—Since AB.BD : BC.BG is compounded of the two ratios AB : BC and BD : BG [V. Def. of compound ratio], the enunciation is the same as if we said, “in the ratio compounded of the ratios of the sides,” which is Euclid’s; but it is more easily understood as we have put it.
Exercises.
1. Triangles which have one angle of one equal or supplemental to one angle of the other, are to one another in the ratio of the rectangles of the sides about those angles.