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.

41

Prop. SO.-If ABC be a circle, AB a diameter, PD a.
fixed line perpendicular to A
AB; then if ACP be any
line cutting the circle in C
and the line PD in P, the
rectangle under AP and AC
is constant. C
Dem.-Since AB is the
diameter of the 0, the /
Z ACB is right (xxxi.); p D.. BCP is right, and BDP
is right;.'. the figure BDPC is a quadrilateral inscribed
in a 0, and, consequently, the rectangle AP. AC
= rectangle AB. AD = constant.
Cor. 1.-This Prop. holds A
true when the line PD cuts the
0, as in the diagram: the value
of the constant will, in this
case, be = AE2. Hence we / D /p
have the following:-
Cor. 2.-If A be the middle c
point of the arc EF, AC any B
chord cutting the line EF in P; then AP. AC = AE2.
On account of its importance, we shall give an independent proof of this Prop. Thus: join EC, and suppose a ( described about the A EPC; then the Z FEA
= ECA, because they stand on equal arcs AF, AE.
Hence AE touches the 0 EPC (xxxii.);;*. the rectangle AP. AC = AE2.
Cor. 3.-If A be a fixed point (see two last figs.),
PD a fixed line, and if any variable point P in PD be
joined to A, and a point C taken on AP, so that the
rectangle AP. AC = constant-say R2-then, by the
converse of this Prop., the locus of the point C is a 0.
DEF.-The point C is called the inverse of the point P,
the 0 ABC the inverse of the line PD, thefixed point A
the centre, and the constant R the radius of inversion.
We shall give more on the subject of inversion in
our addition to Book VYI