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 III.

31

Now, EF. FG = DF. FH (xxxv.), and PF = DPF.
Hence, by addition, making use of II. v., and II. iii.,
EP2 = DF. DI = twice rectangle contained by the
radii.
Prop. 6.-If a circle PGD touch a circle ABC in D
and a chord AB in P, and if c
EF be perpendicular to AB
at its middle point, and at
the side opposite to that of D
the circle PGD, the rectangle
contained by EF and the diameter of thecircle PGDis equal A B
to the rectangle AP. PB. E
Dem.-Let PG be at
right Zs to AB, then PG
is the diameter of the (D
PGD. Join DG, DP, and F
produce them to meet the 0 ABC in C and F; then CF is
the diameter of the 0 ABC, and is 1] to PG (2);.. CF
is 1_ to AB; hence it bisects AB in E (iii.). Through
F draw FH I[ to AB, and produce GP to meet it in H.
Now, since the Z s H and D are right Zs, a semicircle
described on GF will pass through the points D and IH.
Hence HP. PG = FP. PD = AP. PB; (xxxv.)
but HP = EF;.. E.PG = AP. PB.
This Prop. and its Demonstration will hold true when
the Os are external to each other.
Cor. If AB be the diameter of the 0 ABC, this
Prop. reduces to the last.
Prop. 7.-To draw a common tangent to two circles.
Let Pbe the centre E
of the greater 0, Q
the centre of the less, /
with P as centre, and I Ip G
a radius = to the dif- A- c
ference of the radii
of the two Os: describe the 0 IGH;
from Q draw a tangent to this 0, touching it at H.