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.

224

A SEQUEL TO EUCLID.

Prop. 1.-The centres of similitude of an associated
system of figures are concyclic.
Dem.-Let the figures be F1, F2,... F., and I,, Iq, any
two invariable points, A,, Aq the corresponding points
of FA, Fq taken on the lines XI,, XIq. Hence [Sup.
Sect. ii., Prop. 4] the second intersection of the circle
Z with the circumcircle of the triangle XApAq is the
centre of similitude of the figures F,, Fq. Hence the
centre of similitude of each pair of figures of the associated system lies on Z, that is, on the circle through
the invariable points.
DEF. II-.-The circle Z, through the invariable points,
is, on account of the property just proved, called the circle
of similitude of the system.
Prop. 2.-The figure formed by n homologous points is
in perspective with that formed by the invariable points.
This follows from Def. I.
CoR.-Every system of n homologous lines passing
through the invariable points forms a pencil of concurrent
lines.
Prop. 3.-In an associated system of n figures the
points of intersection of n homologous lines are in perspective with the centres of similitude of the figures.
Dem.-Let the homologous be L,, L,,... L,, and
through the invariable points draw lines respectively
parallel to them Then, since these parallels are corresponding lines of F1, F,,... F,,, they are concurrent.
Let them meet in K. Now, consider any two lines
Lp, Lq: the perpendiculars on them from K are respectively equal to their distances from the invariable
points Ip, Iq, and therefore proportional to the perpendiculars on them from the centre of similitude Spq of
the figures Fp, Fq. Hence the point of intersection of
L,, Lq, the point K, and Sa, are collinear, Hence it