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.

104 A SEQUEL TO EUCLID.
points A, B, C, D, and let their diameters be 8, 8', 8",
s"'; then we have
AB. CD BC. AD AC. BD
v_. "/ "+ 'v/ //. 3" v/_v" '8/"'
and by the last Proposition each of the fractions of this
equation remains unaltered by inversion. Hence, if the
diameters of the inverse circles be denoted by d, d, d", d"',
and their common tangents by 12, &c., we get
12. 34 23. 41 13.24
+ _
VdW. 2"d`' Vd/dW. V/'rd ~Vd". V 'd"'
Hence 12. 34 + 23. 14 = 13. 24.'
Cor. 1.-If four arbitrary circles touch a given circle
at a harmonic system of points; then
12. 34 = 23. 14.
Cor. 2.-The theorem of this Proposition may be
written in the form
12. 34 +23. 14 + 1. 24 = 0;
and in this form it proves at once the property of the
"Nine-points Circle." For, taking the Os 1, 2, 3, 4
to be the inscribed and escribed Os of the A, and remembering that when Os touch a line on different
sides, we are, in the application of the foregoing theorem, to use transverse common tangents. Hence,
making use of the results of Prop. 1, Section I., Book
IV., we get
12. 34+23. 14+31. 24
= b2 - 2 + C2 - a2 + a2 - b2 = 0.
Hence the Os 1, 2, 3, 4, are all touched by a fifth 0.
This theorem is due to Feuerbach. The following
simple proof of this now celebrated theorem was pub* This extension of Ptolemy's Theorem first appeared in a
Paper of mine in the Proceedings of the Royal Irish Academy, 1866.