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.

A SYSTEM OF THREE SIMILAR FIGtTRES. 191

IDEF. 2.-The _point K is called the perspective centre
Of the triangle ADADD3
Prop. 2.-In every system of three similar figureo
there is an infinite number of triads of concurrent homologous lines. These lines turn round three 'fixed points
PD, PI, P3 of the circle of similitude, and their point of
concurrence is on the same circle.
Dem.-Let K be the perspective centre of a triangle D, DID, formed by three homologous lines.
Through K draw three parallels, iKP,, KP,, KP, to the
sides of iD1D2ID,. These are three homologous lines.
For ~~(S,, KP2) _(SI, d32) -2
(Si, 1(P3) (Si, d3Q a3'
The point P1 is fixed; for the angle S,KP, is equal to
the inclination KD, to DD,, which is constant. Hence
the arc SP, anad, therefore, the point P1 is given
Similarly the points P2, P3 are fixed.
DEr. 3.-Thie points Pi, P2, P3 are called the invariable
_points, and the triangle iP1P3P3 the invariable triangle.
COR. 1.-The invariable triangle is inversely similar
to the triangle formed by three homologous lines.
For the angle P2P3P1 = PKPI = angle 1D1DAD, and
similarly for the other angles.
Con. 2.-The invariable _points form a system of three
corresponding points.
For the angle
P2SI =a,,andSI,P _ (5,, 1(P,) a,
P2SP3 a~andSP3 -(5,, KP3) a3'
Coia. 3.-The lines of connexion of the invariable points
PI, PI, P3, to any point whatever (K) of the circle of similitude, are three corres~ponding lines of the figures F,,
F3, F.3.