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 FIGURES DIRECTLY SIMILAR. 199
CoR. 2.-The vertices A', B', C' of Brocard'sfirst triangle are the invariable points of the three figures directly
similar, described on the sides of BAC.
For the angle KA'K' is equal to KA"K', and that is
evidently equal to CLe from the properties of the similitude of IBAC, bac; but A'K is parallel to LC. Hence
A'K' is parallel to be. In like manner, B'K', C'K' are
parallel to ac, ab, respectively. Hence A'K', B'K',
C'K', form a system of three corresponding lines, and
A', B', C' are the invariable points.
CoR. 3.-The centre of similitude of the triangles bac,
BAC is a point on the Brocard circle.
For since the figures K'bac, KBAC are similar, and
K'a, KA are corresponding lines of these figures intersecting in A", the centre of similitude [Section ii.,
Prop. 4] is the point of intersection of the circumcircles of the triangles A"aA, A"KK'; but one of
these is the Brocard circle. Hence, &c.
COR. 4.-In like manner, it may be shown that the
centre of similitude of two figures, whose sides are two
triads of corresponding lines of any three figures directly
similar, is a point on the circle of similitude of the three
figures.
CoR. 5. —f three corresponding lines be concurrent,
the locus of their point of concurrence is the BROCARD
CIRCLE.
This theorem, due to M. BROCARD, is a particular
case of the theorem Section Iv., Prop. 2, or of Cor. 1,
due to M'CAY, or of either of my theorems, Cors. 3, 4.
2~. The Nine-points Circle.
6. Let ABC be a triangle, whose altitudes are AA',
BB', CC'; the triangles AB'C', A'BC', A'B'C are inversely similar to ABC. Then if we consider these
triangles as portions of three figures, directly similar,
P2