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.

204

A SEQUEL TO EUCLID.

2. If through the orthocentre of a triangle be drawn any line
L meeting the sides in A', B', C', the lines through A', B', C',
which are the reflections of L, with respect to the sides of the
triangle, are concurrent.
3. If upon the sides of a triangle, ABC, be constructed three
triangles, BCAI, CABi, ABC1, such that A is an excentre of BCA1,
B an excentre of CABi, C an excentre of ABC1,
Prove that(1~) The triangles BCA1, CABi, ABCI, are directly similar.
(2~) A, B, C are the double points.
(3~) The incentres of A1BC, B1CA, C1AB are the invariable
points.
(4~) Al, B1, C1 are the adjoint points.
(5~) The circumcentre of ABC is the director point.
(6) The perspective centre of the triangle formed by three
homologous lines is the orthocentre of that triangle.
(7~) Three homologous lines form a triangle inversely similar
to ABC.
(8~) The lines joining the incentre of A1BC, AB1C, ABCi, are
concurrent.
4. Prove that the triangles, AE'F, DBF', D'EC, fig., p. 183,
are directly similar, and that(1~) The invariable points are the centroids of these triangles.
(2~) The double points are the intersections of the symmedians
of the triangle ABC with the circle through' the invariable
points.
(3~) The director point is the symmedian point of ABC.
(4~) The perspective centre of the triangle formed by any
three homologous lines is the centroid of that triangle.
The application of Tarry's theory of similar figures contained
in this sub-section, with the exception of the theorems 3~ and 5~,
and the Exercises 2 and 4, are due to Neuberg. The demonstration of 5~, given in the text, is nearly the same as one given by
Mr. M'Cay shortly after I communicated the theorem to him.