PROPOSITION 19. THEOREM.
Similar triangles are to one another in the duplicate ratio of their homologous sides.
Let ABC and DEF be similar triangles, having the angle B equal to the angle E, and let AB be to BC as DE is to EF so that the side BC is homologous to the side EF: the triangle ABC shall be to the triangle DEF in the duplicate ratio of BC to EF.
Take BG a third proportional to BG and EF, so that BG may be to EF as EF is to BG; [VI. 11.
and join AG.
Then, because AB is to BC as DE is to EF, [Hypothesis.
therefore, alternately, AB is to DE as C is to EF; [V. 16.
but BC is to EF as EF is to BG; [Construction.
therefore AB is to DE as EF is to BG; [V. 11.
that is, the sides of the triangles ABG and DEF, about their equal angles, are reciprocally proportional;
but triangles which have their sides about two equal angles reciprocally proportional are equal to one another, [VI. 15.
therefore the triangle ABG is equal to the triangle DEF.
And, because BG is to EF as EF is to BG,
therefore BG has to BG the duplicate ratio of that which BC has to EF. [V. Definition 10.
But the triangle ABC is to the triangle ABG as BC is to BG; [VI. 1.
therefore the triangle ABC has to the triangle ABG the duplicate ratio of that which BC has to EF. But the triaingle ABG was shewn equal to the triangle DEF;
therefore the triangle ABC has to the triangle DEF the duplicate ratio of that which BC has to EF. [V. 7.
Wherefore, similar triangles &c. q.e.d.
Corollary. From this it is manifest, that if three
