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.

150 A SEQUEL TO EUCLID.
21. If a variable 0 touch two fixed Os, its radius varies as the
square of the tangent drawn to it from either limiting point.
22. If two Os, whose centres are 0, 0', intersect, as in Euclid
(I. 1), and 00' be joined, and produced to A, and a 0 GDH be
described, touching the Os whose centres are 0, 0', and also
touching the line AO; then, F E
if we draw the radical axis
EE' of the Os, intersecting
00' in C, and the diameter /i 0\
DF of the (0 GHD, and join
EF, the figure CDFE is a A D'D c of B
square.
Dem.-The line joining
the points of contact G and
H will pass through C, the '
internal centre of similitude
of the Os 0, 0'; therefore CG. CH = CE2; but CD2 = CG. CH;
therefore CD = CE.
Again, let 0" be the centre of GDH, and D' the middle point
of AO; then the 0 whose centre is D' and radius D'A touches the
Os O, 0'; hence (by Theorem 7, Section V.) the 1 from 0" on
EE: O"D:: CD': D'A; that is, in the ratio of 2: 1. Hence the
Proposition is proved.
23. If a quadrilateral be circumscribed to a 0, the centre and
the middle points of the diagonals are collinear.
24. If one diagonal of a quadrilateral inscribed in a ( be bisected by the other, the square of the latter = half the sum of the
squares of the sides.
25. If a A given in species moves with its vertices on three
fixed lines, it marks off proportional parts on these lines.
26. Through the point of intersection of two Os draw a line
so that the sum or the difference of the squares of the chords of
the Os shall be given.
27. If two Os touch at A, and BC be any chord of one touching
the other; then the sum or difference of the chords AB, AC bears
to the chord BC a constant ratio. Distinguish the two cases.
28. If ABC be a A inscribed in a 0, and if a 11 to AC through
the pole of AB meet BC in D, then AD is = CD.
29. The centres of the four Os circumscribed about the, As
formed by four right lines are concyclic.
30. Through a given point draw two transversals which shall
intercept given lengths on two given lines.