A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

THEORY OF HARMONIC POLYGONS. 21

217

1 1. If ABC..., A'B'C',... be two homothetic harmonic
polygons of any number of sides; K their homothketic centre;
and if consecutive pairs of the sides of A'B'C'...produced, if necessary, intersect the corr oniy piso
ABC... in the pairs of points cw!, /3Il, yy, &c., the
points aa, -/3/'y', &c., are concyclic.
For, since the figure /3B/3'B' is a parallelogram, BB'
is bisected by /3/3' in B"; and since the ratio of KB:
KB' is given, the ratio of KB:KB" is given, and //'
through B", is parallel to the tangent at B. Hence, &c.
CoRt. I.-If the harmonic polygons of C'or. 2, be quadrilaterals, their circumcircles and that of the octagon
aa'/3/3'7y'88' are coaxal.

For it is easy to see that the squares of the tangents
fromn B to the circuracircies of aa'PP/',yy'88', and A'B'C'D'