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.

74

74 ~~A SEQUEL TO ETUCLlD.

GA2: AB. AC:: 0A2: G'A.GOA:: GA1: G'A:: AD:
AE; but AD -_s -a, anclAE =s;
therefore GA2: AB. AC:: (8 -a): 8.
Cor. 1.- GA' GB2 002
be o ca a b
0A' s -a
For 1
GB2 s- b
In like manner, ca - 8
002 s -0
and - -7
therefore, by addition,
GA 2 GB2 002
tior. 2.-If Of20', 0" 0"' be the centres of the escribed
circles, G'B2 0102 G'A2,
ea ab bc1'&O
Prop. 11.-If r, ]R be the radii of the inscribed and
circumscribed circles of a plane triangle, 6 the distance be.,
tween their centres; then

-+ =1
Dem.-Let 0, P be the centres of the ODs. Join CP, -and
let it meet the circumscribed
O) in D. Join DO, and produce to meet the circumscribed
O) in E. Join EB, GP, PF,
PB, BD). Since P is the centre
of the inscribed 0D, OP bisects
the L ACB;.-. the arc ADA
= the arc DB. ilence the L
ABD = DOB (ILI, 21); and
because PB bisects theL ABC,
theLP BA =PBC;.-.the ZPBI
= DPB;.. DP =DB.

)= POB + P130