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.

162

A SEQUEL TO EUCLID.

Then
AL2: AB. AH:: OL: 00' [vi. Sect. v. Prop. I. (3)].
But AL2 = AB.AE.
Hence AE: AH:: OL: 00';
therefore AE: HE:: OL: O'L.
In like manner GE: BE:: OL: O'L.
Hence AG: BH:: OL: O'L,

that is, in a given ratio. Therefore the tangents AK, BK are in
a given ratio [EUCLID, VI. iv. Ex. 2]; and the locus of K is a
circle coaxal with O, 0'.
This theorem is the reciprocal of a remarkable one in Confocal
Conics (see Conics, page 184). The demonstration of it here
given, as well as that of the Proposition Ex. 116, have been communicated to me by W. S. M'CAY, F.T.C.D.
123. In the same case the locus of the point E is a circle coaxal with O, 0'.
124. If O" be the centre of the locus of E, then LO" is half
the harmonic mean between LO and LO'.