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.

BOOK VI.

107

showing that the four circles are all touched by a circle
having the circle 4 on one side, and the other three
circles on the other. This proof of Dr. Hart's extension of Feuerbach's theorem was published by me in
the Proceedings of the Royal Irish Academy in the year
1866.
Prop. 12.-If two circles X, Y be so related that a
triangle may be inscribed in X and described about Y, the
inverse of X with respect to Y is the " Nine-points Circle"
of the triangle formed by joining the points of contact
on Y.
Dem.-Let ABC be the A
inscribed in X and described
about Y; and A'B'C' the A / D F
formed by joining the points
of contact on Y./
Let O, O' be the centres E
of X and Y. Join O'A, inter- B A'
secting B'C' in D; then, evidently, D is the inverse of
the point A with respect to
Y, and D is the middle point of B'C'. In like manner,
the inverses of the points B and C are the middle
points C'A' and A'B';.. the inverse of the ( X, which
passes through the points A, B, C with respect to Y,
is the ( which passes through the middle points of
B'C', C'A', A'B', that is the "Nine-points Circle" of the
triangle A'B'C'.
Cor. 1.-If two Os X, Y be so related that a A inscribed in X may be described about Y, the 0 inscribed in the A, formed by joining the points on Y,
touches a fixed circle, namely, the inverse of X with
respect to Y.
Cor. 2.-In the same case, if tangents be drawn to
X at the points A, B, C, forming a new A A"B"C",
the 0 described about A"B"C" touches a fixed circle.
Cor. 3. —Join 00', and produce to meet the ( X in
the points E and F, and let it meet the inverse of X
with respect to Y in the points P and Q; then PQ is
the diameter of the "Nine-points Circle" of the A