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. 197

Dem. —Join DF. Then since AF = KE'; but KE'
=DC';.'. AF = DC'. Hence [Euc. I. xxxiv.] DF is
parallel to AC'-that is, to AO, &c.
In the same manner it may be proved that the sides
of D'E'F are parallel to AM', BM', CQ', respectively.
CoR. 3.-The six sides of Lemoine's hexagon, taken in
order, are proportional to sin (A- )), sin o, sin (B - w),
sin w, sin (C - w), sin o.
COR. 4. — and K are the Brocard points of the triangle DEF, and 2' and K of D'E'F'.
CoR. 5.-The lines AA', BB', CC' are isogonal conjugates of the lines Ap, Bf, Cr (SECTION II., PROP. 2,
Cor. 2) with respect to the triangle ABC.
DEF.-If the Brocard circle of the triangle ABC meet
its symmedian lines in the points A", B", C", respectively,
A"B"C" is called Brocard's second triangle.
Prop. 4.-Brocard's second triangle is the triangle of
similitude of threefigures, directly similar, described on
the three sides of the triangle ABC.
Dem.-Since OK (fig., Prop. 5) is the diameter of the
Brocard circle, the angle OA"K is right. Hence A"
is the middle point of the symmedian chord AT, and is
therefore [Section II., Prop. 4, Cors. 1, 5] the double
point of figures directly similar, described on the lines
BA, AC. Hence [Section iv., Def. 1] the proposition
is proved.
Prop. 5.-Iffigures directly similar be described on the
sides of the triangle ABC, the symmedian lines of the triangle formed by three corresponding lines pass through
the vertices of Brocard's second triangle.
Dem.-Let bac be a triangle formed by three corresponding lines, then bac is equiangular to BAC; and
since A" is the double point of figures described on BA,
AC, and ba, ac are corresponding lines in these figures,
the line A"a divides the angle bac into parts respecP