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.

190

A SEQUELrTO EUCLID.

Dem.-Let dl, d2, d3 be three homologous lines
forming the triangle DD2D3; we have, by hypothesis,
(SI, d2) _2 (2, 43) _ a3 (S3, di) a,
(Si, d3) -3 (2, d,) a, (83, d2) a-
Hence it follows that the lines SID,, S2D2, S3D3 cointersect in a point K, whose distances from the lines
d,, d2, d are proportional to a,, a3, a3. The triangle DD2D3
being given in species, its angles are the supplements
of a,, a2, as. Hence the angles DKD2, DKD3, D3KD,,

As

are constants; that is, the angles SIKS2, S2KS3, S3KS1
are constants. Hence the point K moves on three
circles passing through Si and S, S2 and S3, S3 and S;
that is, it moves on the circumcircle of the triangle
1S8S33.