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.

178 A SEQUEL TO EUCLID.
COR. 2.-The three angles tAB, 2BC, 2CA are equal
to one another, and each is equal to the Brocard angle of
the triangle.
Dem.-The angles are equal [EEuc. III. xxxII.]. Let
their common value be w. Then, since the lines AU,
B]3, C are concurrent, we have, from Trigonometry,
sin-3 = sin (A - )) sin (B - o) sin (C - c).
Hence cot o = cot A + cot B + cot C.
Therefore o is the Brocard angle of the triangle (Section i., Prop. 8).
Exercises.
1. Inscribe in a given triangle ABC a rectangle similar to a
given rectangle, and having one side on the side BC of the triangle. A is the homothetic centre of the sought rectangle, and a
similar rectangle constructed on the side BC.
2. Inscribe in a given triangle a triangle whose sides will be
parallel to the three given lines.
3. From the fact that a triangle ABC, and the triangle A'B'C',
whose vertices are the middle points of the sides of ABC, are
homothetic; prove-1~, that the medians of ABC are concurrent;
2~, that the orthocentre, the circumcentre, and the centroid of
ABC are collinear.
4. Show that Proposition 9 of Book VI., Section I., and its
Cor. are applications of the theory of figures directly similar.
5. If figures directly similar be described on the perpendiculars of a triangle, prove that their double points are the feet of
perpendiculars let fall from the orthocentre on the medians.
6. The Brocard points are isogonal conjugates with respect to
the triangle BAC.
7. The system of multiples for which f is the mean centre of
1 1 1 1 11
A, B, C is b, -, -; and the system for n' is - -,.
8. If the line A'B' (Fig., Prop. 4) turn round any given point
in the plane, while AB remains fixed, the locus of the double
point 0 is a circle.