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.

186 A SEQUEL TO EUCLID.
centre of T. Therefore the centre of T is the incentre
of the triangle afty. Similarly, the excentres of apy
are the centres of T,, T,, T3.
CoR. 1.-Taylor's circle T is one of the Tucker system
of the triangle ABC.
For, if we consider the triangle KK"N' inscribed in
ABC, the angle KK"N' is equal to KK'N', since the
points K, K', K", N are concyclic; but KK'N' is equal
to C, since K'N' is antiparallel to AC. Hence KK"N
is equal to C. Again, N'KK" is equal to N'K'K",
which, since K'K" is parallel to BC, is equal to K'N'B,
and therefore equal to A. Therefore KK"N' is similar
to ABC. Hence its circumcircle T is one of the
Tucker system of the triangle ABC.
COR. 2.-The radical axes of the circles T, T,, T,, T3
taken in pairs are the sides and the altitudes of the triangle ABC.
COR. 3.-The figureformed by the centres of T, Tj, T,,
T3 is similar to, and in perspective woith, that formed by
the points H, A, B, C.
For, H, A, B, C are the incentre and the excentres
of the triangle A'B' C', which is similar to, and in perspective with, afly.
Prop. 7.-Taylor's circle T cuts orthogonally the
three escribed circles of the orthocentric triangle of ABC,
and each of the circles T,, T,, T, cuts orthogonally the
inscribed and two of the escribed circles of the same triangle.
Dem.-Let the perpendiculars from A, B, C on the
lines B'C', C'A', A'B', respectively, be 7rl,, 7r3, respectively; then r,, 7r,, T3 are the radii of the escribed
circles of A'B'C'. Now, since the triangles AB'C',
ABC are similar, 7r-2: AA':: AC'2: AC2; that is,,2:
AN. AC:: AC. AK": AC2;.'. 1r2= AN. AK"; but
AN. AK" is equal to the square of the tangent from
A to T. Hence the circle whose centre is A, and radius