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.

THEORY OF FIGURES DIRECTLY SIMILAR. 203
R the circumradius, we have 2r2 = AI. II', and
21r = AI. ID. Hence r:::: II': DI.
Again, through I draw IF parallel to EA". Now,
since the points I', A", in the triangle AB'C', correspond to I and 0 in ABC, the angle AI'A"= AIO.
Hence the angle II'F is equal to DIO, and the angle
I'IF is equal to IDO, because each is equal to DAO.
Hence the triangles II'F and DIO are equiangular.
Therefore II': DI:: IF: DO. Hence IF: DO:: r: R.
Therefore IF = r. Now since EA" and IF are parallel, and are radii respectively of the nine-points
circle, and incircle of ABC, the line FA" passes through
their centre of similitude. Hence the proposition is
proved.
Similarly, if J' be the centre of any of the escribed
circles of the triangle AB'C', the line A"J' passes through
the point of contact of the nine-points circle of ABC with
the corresponding escribed circle.
Exercises.
1. If Al, Bi, C, be the reflexions of the angular points A, B, C
of the triangle ABC, with respect to the opposite sides, then the
triangles AiBC, AB1C, ABCi, being considered as portions of
three figures directly similar,
Prove that(1~) A, B, C are the double points.
(2~) The orthocentres of AiBC, ABIC, ABC1, are the invariable points.
(3~) Ai, B1, C, are the adjoint points.
(4~) The orthocentre of ABC is the director point.
(5~) The incentre of the triangle formed by three homologous
lines is its perspective centre.
(6~) The triangle formed by any three homologous lines is
similar to the orthocentric triangle of ABC.
(7~) The lines joining the orthocentres of AiBC, AB1C, ABC1
to their incentres are concurrent.