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.

140 A SEQUEL TO EUCLID.
Exercises,
1. The eight circles which touch three given circles may be
divided into two tetrads-say X, Y, Z, W; X', Y, Z', W'-of
which one is the inverse of the other with respect to the circle
cutting the three given circles orthogonally.
2. Any two circles of the first tetrad, and the two corresponding circles of the second, have a common tangential circle.
3. Any three circles of either tetrad, and the non-corresponding
circle of the other tetrad, have a common tangential circle.
4. Prove by means of the extension of Ptolemy's Theorem
(the middle points of the sides being regarded as very small
circles) that these point-circles, and the inscribed circle, or any
of the escribed circles, have a common tangential circle.
5. The anharmonic ratios of the four points of contact of the
"nine-points circle" with the inscribed and the escribed circles
are respectively
a2 _ b2 b2 _ 2 c2 _ a2
a" - c-' bi- a.' c2 - b2