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.

A SYSTEM OF THREE SIMILAR FIGURES. 193

The triangle 'S/2'S3' is in perspective both with the
invariable triangle and with the triangle of similitude;
and the three triangles have a common centre of perspective.
Dem. —By hypothesis, the three points S1', S1, S,
are homologous points of the figures Fi, F2, F3. Hence
the lines S'/P, S,1P, S1P3 are concurrent. Hence the
points S1', P1, 81 are collinear. Similarly S,', P,, S2
are collinear, and S3', P3, S3 are collinear. Hence the
proposition is proved.
DEF. 5. —The points S1, S2', S3' are called the adjoint
points of the figures.
Prop. 6. —In three figures, F1, F,, F3, directly similar,
there exists an infinite number of systems of three corresponding points which are collinear. Their loci are three
circles, each passing through two double points and
through E, the centre of perspective of the triangle of
similitude, and the invariable triangle. Also the line
of collinearity of each triad of corresponding points passes
through E.
Dem.-Let C, C2, C3 be three homologous collinear
points. Since S2 is the double point of the figures F3,
F1; the triangles S2C3C1, S2P3P, are similar; therefore
the angle S2C3C1 is equal to the angle S2P3P1, and
therefore [EEuc. III., xxi.] equal to the angle SSE.
In like manner, the angle C2C3S1 is equal to SS2E;
therefore the angle S2C381 is equal to S2ES,. Hence
the locus of C3 is the circumcircle of the triangle SIES2.
Again, since S2C3ES1 is a cyclic quadrilateral, the
angles SSE, EC3S2 are supplemental. Hence the
angles S2C3C1, EC3S2 are supplemental; therefore the
points C1, C3, E are collinear, and the proposition is
proved.
COR. 1. —The circumcircle of the triangle SES3 passes
through Ss'. For S3' is a particular position of C3.