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.

LEMOINE'S, TUCKER'S, AND TAYLOR'S CIRCLES. 185
the Taylor circle of the triangle. I shall denote it by
the letter T, and the Taylor circles of the triangles
BHC, CHA, AHB, respectively, by T1, T2, T3.
Prop. 5.- The chords KN, I'N', K"N" of T meet the
sides of the triangles BHC, CHA, AHB, respectively, in
their points of intersection with the Taylor's circles of
these triangles.
Dem.-Let KN meet BH in L, and HC in M. Now
it is evident that the point A' is common to the circumcircles of the four triangles formed by the lines AB,
AC, BB', CC'. Hence [Book III., Prop. 12, Cor. 2]
the projections of A on these lines are collinear; therefore the points L, M are the projections of A' on BH,
HC, respectively. Similarly MI' is the projection of
B' on HC, and M" of C' on BH. Therefore the circle
T, passes through the points L, IM; M', N'; MI", N";
that is, through the points of intersection of KN, K'N',
K"N", with the sides of the triangle BHC. Hence the
proposition is proved.
Prop. 6.-The centresof T, T1, T2, T coinciderespectively
with the incentre and the excentres of the triangle formed
by joining the middle points of the sides of the orthocentric
triangle of ABC.
Dem.-The line KN is evidently the Simson's line
of the point A' with respect to the triangle BHC', and
C' is the orthocentre of BHC'. Hence A'C' is bisected
by KN [Book III., Prop. 14]. Similarly, KN bisects
A'B', therefore it bisects two of the sides of the triangle
A'B'C', and similar properties hold for K'N', and K"N.
Hence, if a, /3, y be the middle points of the sides of
A'B'C', each of the lines KN, K'N', K"N"passes through
two of these points. Again, since B'C' is bisected in
a, the triangle aN'N" is isosceles, and the bisector of
the angle a bisects N'N" perpendicularly, and therefore
passes through the centre of T. Similarly, the bisectors
of the other angles of the triangle agy pass through the