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. 201

Third triad of corresponding lines; perpendiculars
at the middle points of the corresponding lines C'A,
C'A', CA'.
The point of concurrence of these triads are the
middle points A"', B"', C"' of the sides of ABC.
The point of concurrence of the lines of F1 of these
triads is the middle A" of AH; the point of concurrence of the lines of F2 is the middle B" of BH; and
of the lines of F3 the middle C" of CH.
The points A", B", C" are the invariable points.
Hence the nine points, viz., A', B', C' (centres of similitude); A", B", C" (invariable points); A"', B"', C"'
(points of concurrence of triads of corresponding lines),
are on the circle of similitude. Hence the circle of
similitude is the nine-points circle of the triangle.
Hence we have the following theorems:1~. Three homologous lines of the triangles AB'C',
A'BC', A'B'C form a triangle afly in perspective with
A'B'C'; the centre of perspective, N, is on the ninepoints circle of ABC, and it is the circumcentre of a/3y.
for its distances to the sides of a/3y are:: cos A: cos B: cos C. For example, the Brocard lines of the three
triangles possess this property.
2~. Lines joining the points A", B", C" to three homologous points F1, F2, F3 are concurrent, and meet on the
nine-points circle of ABC.
3~. If P, PI, P2, P3 be corresponding points of the triangles ABC, AB'C', A'BC', A'B'C, the lines A"P1,
B"P,, C"P3 meet the nine-points circle of ABC in the
point which is the isogonal conjugate with respect to the
triangle A"B"C" of the point of infinity on the line
joining P to the circumcentre of ABC.
4~. Every line passing through the orthocentre H meets
the circumcircles of the triangles AB'C', A'BC', A'B'C in
corresponding points.
5~. The lines joining the points A", B", C" to the
centres of the inscribed circles of the triangles AB'C',