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.

221

symmedian of the triangle A'B'C'. Similarly, B'B, C'C are
symmedians.
2. The centres of inversion of a harmonic polygon are the
limiting points of its circumcircle and Brocard circle, and the
Lemoine line is their radical axis.
3. The product of any two alternate sides of a harmonic polygon is proportional to the product of the sines of their inclinations
to their included side.
For if A, B, C, D be four consecutive vertices, A', B', C', D' the
corresponding vertices of a regular polygon, the anharmonic
ratio (ABCD) = (A'B'C'D'). Hence (AB. CD) / (AC. BD)=
(A'B'. C'D') / (A'C'. B'D') = (sec2 i/n) / 4; but AC = 2 R sin
ABC, BD = 2R sin BCD.
Hence (AB. CD)/sin ABC. sin BCD = R2 sec2 7r/n.
4. If we invert the sides A'B' (see fig. p. 207) of a regular
polygon with respect to S, we get a circle passing through AB
and S. Hence, if through the extremities of each side of a
harmonic polygon circles be described passing through either of
the centres of inversion, these circles cut the circumcircle at a
constant angle -7r/n.
5. In the same case they all touch another circle, and the
points of contact are the vertices of a harmonic polygon.
6. If through the symmedian point, and any two adjacent
vertices of a harmonic polygon, a circle be described, it cuts the
circumcircle at a constant angle.
7. A system of circles passing through the two centres of
inversion of a harmonic polygon, and passing respectively
through its vertices, cut each other at equal angles, and cut its
circumcircle and Brocard circle orthogonally.
8. In the same case the points of intersection on the Brocard
circle are the vertices of a harmonic polygon.
9. Prove that the centre of similitude of the two polygons
formed by the alternate vertices in Prop. 9 is the symmedian
point of the original polygon, and that the centre of similitude of
either, and the original polygon, is a Brocard point of the
original polygon.
10. Prove that the circles in Ex. 5, described through the symmedian point, and through adjacent vertices of a harmonic polygon, all touch a circle coaxal with the Brocard circle and the
circumcircle, and that the points of contact are the vertices of a
harmonic polygon,