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.

218

A SEQUEL TO EUCLID.

are in the ratio 1: 2; and the squares of tangents from
C, D, A to the same circles are in the same ratio.
CoR. 2.-If the two harmonic polygons of Cor. 2 have
an even number of sides, the n points of intersection of the
sides of the first with the corresponding opposite sides of
the second, respectively, are collinear.
Dem.-For simplicity, suppose the figures are quadrilaterals, but the proof is general. Let P be the point
of intersection; then the angle ABE = AC'D'. Hence
ABC'D' is a cyclic quadrilateral. Therefore P is a
point on the radical axis of the circumcircles. Hence
the proposition is proved.
COR. 3.-In the general case the lines aa', /33', yy' are
the sides of a polygon, homothetic with that formed by the
tangents at the angular points A, B, C, &c. Hence it
follows, if the harmonic polygon ABC... be of an even
number of sides, that the intersections of the lines aa',
/3 ', y'7, taken in opposite pairs, are collinear.
Prop. 12.-The perpendiculars from the circumcentre
of a harmonic polygon, of any number of sides n on the sides,
meet its Brocard circle in n points, which connect concurrently in two ways with the vertices of the polygon.
This general proposition may be proved exactly in
the same way as Prop. 2, page 196.
DEF.-If the points of concurrence of the lines in this
proposition be 0, f', these are called the BRocARD POINTS
of thepolygon; and the n points L, M, N, &c., in which
the perpendiculars meet the Brocard circle, for the same
reasons as in Cor. 2, p. 194, are called its INVARIABLE
POINTS. Also the points of bisection of the symmedian
chords AK, BK, CK, &c., will be its DOUBLE POINTS.
CoR. 1.-The n linesjoining respectively the invariable
points L, M, N... to n corresponding points offigures
directly similar described on the sides of the harmonic