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.

206

A SEQUEL TO EUCLID.

Hence, if the inverses of the systems of points A, B, C... A', B', C'... be the systems a/3y,... a'^...,
the lines aa', /3/', yy'... are concurrent. Let their
common point be K; and since, evidently, the points
A, B, C, B' form a harmonic system, their inverses, the
points a, /3, y, 3', form a harmonic system; but the
line //3' passes through K. Therefore the perpendiculars from K on the lines a/3, 3y are proportional to
these lines. Hence the proposition is proved.
CoR. 1.-If the vertices of a harmonic polygon of n
sides be 1, 2, 3... n, and K its symmedian point, the
re-entrant polygon formed by the chords 13, 24, 35, &c.,
is a harmonic polygon, and K is its symmedian point.
This is proved by showing that the perpendiculars
from K on these chords are proportional to the chords.
Thus, let A, B, C be any three consecutive vertices;
p, p' perpendiculars from K on the lines AB, AC; and
let AK produced meet the circumcircle again in A;
then it is easy to see that the ratio P: - is equalto
AB' AC
the anharmonic ratio (ABCA'), which is constant, because [Book VI., Section iv., Prop. 9] it is equal to
the corresponding anharmonic ratio in a regular polygon,
and is constant. Hence A- is constant.
AB AC
CoE. 2.-In the same manner the polygon formed by
the chords 14, 25, 36 is a harmonic polygon, and Kis its
symmedian point, &c.
COR. 3.-The vertices of any triangle may be considered
as the inverses of the angular points of an equilateral triangle.
COR. 4.-A harmonic quadrilateral is the inverse of a
square; and its symmedian point is the intersection of its
diagonals.