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.

106

A SEQUEL TO EUCLID.

nearity evidently is I to AD and it bisects PD (see
Prop. 14, Section I., Book III.). Hence FG is the line
of collinearity, and FG is I to AD. Let M be the
point of contact of 0 with BC; join GM, and let fall
the L HS. Now, since FM is a tangent to 0, if from
N we draw another tangent to 0, we have FM2 = FN2
+ square of tangent from N (Prop. 21, Section I.,
Book III.); but FM = ' (AB - AC). Hence FM2
= FR. FI (Prop. 8, Cor. 5, Section I., Book IV.)
= FK. FN;.-. square of tangent from N = FN. NK.
Again, let GT be the tangent from G to 0; then GT2
= square of tangent from N + GN2 = FN. NK + GN2
= GF2. Hence the 0 whose centre is G and radius
GF will cut the circle 0 orthogonally; and.'. that (D
will invert the circle 0 into itself, and the same 0
will invert the line BC into; and since BC touches 0,
their inverses will touch (Prop. 2). Hence E touches
0, and it is evident that S is the point of contact.
In like manner, if M' be the point of contact of O'
with BC, and if we join GM', and let fall the 1 HS'
on G1M', S' will be the point of contact of X with O'.
Cor.-The circle on FR as diameter cuts the circles
0, 0' orthogonally.
Prop. 11. —Dto HAET'S EXTENSION OF FEUERBACcH'S
THEOREi:-If the tFhree sides of a plane triangle be replaced by three circles, then the circles touching these, which
correspond to the inscribed and escribed circles of a plane
triangle, are all touched by another circle.
Dem.-Let the direct common tangents be denoted,
as in Prop. 11, by 12, &c., and the transverse by 12',
&c., and supposing the signs to correspond to a A whose
sides are in order of magnitude a, b, c; then we have,
because the side a is touched by the ( 1 on one side,
and by the O(s 2, 3, 4 on the other side,
12'. 34 14'. 23 = 13'. 24;
12'. 34 + 24'. 13 = 23'. 14;
13'. 24 + 234'. 12= 23'. 14.
Hence 14' - 23 + 34'. 12 = 24'. 13;