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.

SECTION III.
THEORY OP IHARMONIC SECTION,
DEF. - If a line AB be divided internally in the
point C, and ex- A O C B D
ternally in the '- ' — ' I
point D, so that the ratio AC: CB =- ratio AD: DB;
the points C and D are called harmonic conjugates to
the points A, B.
Since the segments AC, CB are measured in the same
direction, the ratio AC: CB is positive; and AD, DB
being measured in opposite directions, their ratio is
negative. This explains why we say AC: CB = - AD: DB. We shall, however, usually omit the sign minus,
unless when there is special reason for retaining it.
Cor.-The centres of similitude of two given circles
are harmonic conjugates, with respect to their centres.
Prop. 1.-If C and D be harmonic conjugates to A and
B, and if AB be bisected in 0, then OB is a geometric
mean between OC and OD.
Dem.- AC: CB:: AD: DB;
AC-CB AC+CB AD- DB AD+DB
2 2 2 2
or OC: OB:: OB: OD.
Hence OB is a geometric mean between OC and OD.
Prop. 2.-If C and D be harmonic conjugates to A and
B, the circles described on AB and CD as diameters intersect each other orthogonally.
Dem. —Let the Os intersect in P, bisect AB in O; A C B D
join OP; then, by Prop. 2,
we have OC. OD = OB2 = OP2.
Hence OP is a tangent to the
circle CPD, and therefore the Os cut orthogonally.