Dem.—Since the figures A and C are similar, they are equiangular, and have the sides about their equal angles proportional. In like manner B and C are equiangular, and have the sides about their equal angles proportional. Hence A and B are equiangular, and have the sides about their equal angles proportional. Therefore they are similar.
Cor.—Two similar rectilineal figures which are homothetic to a third are homothetic to one another.
Exercise.
If three similar rectilineal figures be homothetic, two by two, their three centres of similitudes are collinear.
PROP. XXII—Theorem.
If four lines (AB, CD, EF, GH) be proportional, and any pair of similar rectilineal figures (ABK, CDL) be similarly described on the first and second, and also any pair (EI, GJ) on the third and fourth, these figures are proportional. Conversely, if any rectilineal figure described on the first of four right lines: the similar and similarly described figure described on the second :: any rectilineal figure on the third : the similar and similarly described figure on the fourth, the four lines are proportional.
Dem. 1.—ABK : CDL :: AB2 : CD2. [xx.];
| and | EI | : GJ :: EF2 : GH2 [xx.]. | |||||||||
| But since | AB | : CD :: EF : GH, | |||||||||
| AB2 | : CD2 :: EF2 : GH2 [V. xxii., Cor. 1]; | ||||||||||
| therefore | ABK | : CDL :: EI : GJ. |