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.

BOOK VI.

123

The foregoing construction holds for each case, the
first of which admits of four solutions, and the second
of two.
Cor. 3.-Similarly, we may suppose one of the circles
to open out into a line, and we have the problem: " To
describe a circle touching a line and two given circles";
and if two circles open out into lines, the problem:
" To describe a circle touching two given lines and a circle."
The foregoing construction extends to these cases also,
and like observations apply to the remaining cases,
namely, when one of the circles reduces to a point, and
one opens out into a line, &c. Since our construction
embraces all cases, except where the three circles become three points or open out into three lines, it would
appear to be the most general construction yet given
for the solution of this celebrated problem.
Another Method-Analysis.-Let 0, 0', O' be the
centres of the Os X, Y, Z, and let AR, Bit be the
radical axis of the pairs of
Os XY, YZ, respectively,
and let O"' be the centre of
the required 0 W: from
0"' let fall the Is O"'A,
O"'B; join R to C, the w
point of contact of W with A
Z, and produce it to meetB
O"D drawn I to O"'R. Now,
because W touches the Os \ o
X, Y, its radius O"'C has a
givenratio to O"'A (Prop.7).
Similarly, O"C has a given D
ratio O"'B;.-. 0"'A has a
given ratio to O"'B; hence
the line O"'R is given in position, and the ratio of
O"'R: 0'"B3 is given;.. the ratio of O"': O"'.0 is
given; hence the ratio of O"D: O"C is given;.*. D is
a given point and R is a given point;.'. the line RD is
given in position; hence C is a given point. Similarly,
the other points of contact are given,