5. If two figures be directly similar, and have a pair of homologous sides parallel, every pair of homologous sides will be parallel.
Def. iii.—Two figures, such as those in 5, are said to be homothetic.
6. If two figures be homothetic, the lines joining corresponding angular points are concurrent, and the point of concurrence is the centre of similitude of the figures.
7. If two polygons be directly similar, either may be turned round their centre of similitude until they become homothetic, and this may be done in two different ways.
8. Two circles are similar figures.
Dem.—Let O, O′ be their centres; let the angle AOB be indefinitely small, so that the arc AB may be regarded as a right line; make the angle A′O′B′ equal to AOB; then the triangles AOB, A′O′B′ are similar.
Again, make the angle BOC indefinitely small, and make B′O′C′ equal to it; the triangles BOC, B′O′C′ are similar. Proceeding in this way, we see that the circles can be divided into the same number of similar elementary triangles. Hence the circles are similar figures.
9. Sectors of circles having equal central angles are similar figures.
10. As any two points of two circles may be regarded as homologous, two circles have in consequence an infinite number of centres of similitude; their locus is the circle, whose diameter is the line joining the two points for which the two circles are homothetic.