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.

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SECTION VI.
THEORY OF HARMONIC POLYGONS.
DEF. I.-A cyclic polygon of any number of sides, having a point K in its plane, such that perpendiculars from
it on the sides are proportional to the sides, is called a
harmonic polygon.
DEF. I.-The point K is called the symmedian point
of the polygon.
DEF. IIr.-The lines drawn from K to the angular
points of the polygon are called its symmedian lines.
DEF. IV.-Two figures having the same symmedian
lines are called co-symmedian figures.
DEF. v.-If 0 be the circumcentre of the polygon, the
circle on OK, as diameter, is called its Brocard circle.
DEF. VI.-If the sides of the polygon be denoted by
a, b, c, d,... and the perpendiculars on them from K
by x, y, s, u,... then the angle o, determined by any
of the equations x = -i a tan o, y = - b tan o, &c., is called
the Brocard angle of the polygon.
Prop. 1.-The inverses of the angular points of a
regular polygon of any number of sides, with respect to
any arbitrary point, form the angular points of a harmonic polygon of the same number of sides.
Dem.-Let A, 13, C... be the angular points of the
regular polygon; A', B', C'... the points diametrically
opposite to them. Now, inverting from any arbitrary
point, the circumcircle of the regular polygon will
invert into a circle X, and its diameters AA', BB',
CC'... into a coaxal system Y, Y1, Y2, &c; then
[VI., Section v., Prop. 4] the radical axes of the pairs
of circles X, Y; X, Y1; X, Y2, &c., are concurrent.