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.

192

A SEQUEL TO EUCLID.

In fact these lines pass through three homologous
points, P,, P2, P3, and make with each other, two by
two, angles equal to a,, a2, a%.
Prop. 3. — The triangle formed by any three corresponding points is in perspective with the invariable triangle,
and the locus of their centre of perspective is the circle of
similitude.
Dem.-Let B1, B1, B3 be three corresponding points;
then PB,, P2B2, P3B3 are three corresponding lines;
and since they pass through the invariable points they
are concurrent, and their point of concurrence is on
the circle of similitude. Hence the proposition is
proved.
Prop. 4. —The invariable triangle and the triangle of
similitude are in perspective, and the distances of their
centre of perspective from the sides of the invariable triangle are inversely proportional to al, a2, a,.
Dem. —We have
a2 S1,P (S1, PIP2) a, (S, P2P3) a, (S3, P3P1)
a3 S1P3 (S1, PIP3) a, - (S2, P2PI) a2 (S3, P3P2)
Hence the lines S1,P, S2P,, S3P3 are concurrent.
DEF. 4.-The centre of perspective of the invariable
triangle, and the triangle of similitude, is called the
director point of the three similar figures F,, F, F3.
Prop. 5. —
Let S1' be the point of F, which is homologous to S1,
considered in F2 and F3.
Let S2' be the point of F2, which is homologous to S2,
considered in F3 and F,.
Let S3' be the point of F3, which is homologous to S,,
considered in F1 and F,.