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.

30

A SEQUEL TO EUCLID.

following important theorem. The lines drawn from
the point of contact of two touching circles to the extremities of any diameter of one of them, will meet
the other in points which will be the extremities of a
parallel diameter.
Prop. 3. —If two circles touch at P, and any line PAB
cut both circles in A and B, the tangents at A and B are
parallel.
Dem. —Let the tangents at A and B meet the tangents at P in the points E and F.
Now, since AE = EP (1), the Z APE = PAE. In
like manner, the Z BPF =PBF;.. Z PAE = PBF,
and AE is 1I to BF.
This Prop. may be inferred from (2), by supposing
the lines PAB, PCD to approach each other indefinitely;
then AC and BD will be tangents.
Prop. 4.-If two circles touch each other at any point
P, and any line cut the circles in
the points A, B, C, D; then the -
angle APB = CPD.
Dem.-Draw a tangent PE
at P; then A D
Z EPB = PCB; (xxxii.)
Z EPA = PDA.
Hence, by subtraction, Z APB
= CPD.
Prop. 5.-If a circle touch a semicircle in D and its
diameter in P, and PE be per- E
pendicular to the diameter at P,
the square on PE is equal to
twice the rectangle contained
by the radii of the circles.
Dem.-Complete the circle, A - B
and produce EP to meet it
again in G. Let C and F be the
centres; then the line CF will
pass through D. Let it meet G - H
the outside circle again in H.