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 VI.

89

Hence the reciprocals of lines in arithmetical progression
are in harmonical progression.
Prop. 4. —Any line cutting a circle, and passing through
a fixed point, is cut harmonically by the circle, the point,
and the polar of the point.
Let D be the point, EF its polar, DGH a line cutting the ( in the points G and H, and the polar of D
in the point J; then the M
points J, D will be har-
monic conjugates to 1: and
G.
Dem.-Let 0 be the cen- c| D
tre of the 0; from 0 let;
fall the.L OK on HD; then,
since K and C are right Z s,
OKJC is a quadrilateral in a
O;.. OD.DC=KD.DJ; butOD. D=DE2;.-. KD.DJ
= DE. Hence KD: DE:: DE: DJ; and since KD,
DE are respectively the arithmetic mean and the
geometric mean between DG and DH, DJ (Prop. 3.)
will be the harmonic mean between DG and DI.
The following is the proof usually given of this
Proposition:-Join OH, OG, CH, CG. Now OD. DC
= DE2 = DH. DG;.'. the quadrilateral HOCG is inscribed in a (;.'. the angle OCH = OGH; and DCG
= OHD; but OGH = OHD;.-. OCH = DCG. Hence
HCJ = GCJ; hence CJ and CD are the internal and
external bisectors of the vertical angle GCH of the
triangle GCH; therefore the points J and D are harmonic conjugates to the points H and G. Q. E. D.
Cor. 1.-If through a fixed point D any line be
drawn cutting the ( in the points G and H, and if
DJ be a harmonic mean between DG and DH, the
locus of J is the polar of D.
Cor. 2.-In the same case, if DK be the arithmetic
mean between DG and DH, the locus of K is a 0,
namely, the ( described on OD as diameter, for the
4 OKD is right,