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.

196

A SEQUEL TO EUCLID.

to ACB. In like manner, the other angles of these
triangles are equal; and since they have different
aspects, they are inversely similar.
CoR.-The three lines A'K, B'K, C'K, produced, coincide with Lemoine's parallels. For since the angle O'AK
is right, A'K is parallel to B C.
Prop 2.-The three lines A'B, B'C, C'A are concurrent, and meet on the Brocard circle, in one of the Brocard points.
Dem.-Produce BA', CB' to meet in 0. Then since
the perpendiculars from K, on the sides of ABC, are
proportional to the sides, and these perpendiculars are
equal, respectively, to A'X,B'Y, C'Z, the triangles BA'X,
CB'Y, A'CZ are equiangular;.*. the angle BA'Xis equal
to CB'Y, or [JEuc. I. xv.] equal to 2B'O. Hence the
points A', Q, B', 0 are concyclic, and.-. BA', CB' meet
on the Brocard circle. In like manner, BA', AC' meet
on the Brocard circle. Hence the lines A'B, B'C, C'A
are concurrent, and evidently (Section ii., Prop. 5) the
point of concurrence is a Brocard point. In the same
manner it may be proved that the three lines AB', BC',
C'A meet on the Brocard circle in the other Brocard
point.
Prop. 3.-The lines AA', BB', CC' are concurrent.
Dem —Since Lemoine's circle, which passes through
F' and E, and Brocard's circle, which passes through
A' and K, are concentric, the intercept F'A' is equal to
KE. Hence the lines AA', AK are isotomic conjugates
with respect to the angle A. In like manner, BB', BK
are isotomic conjugates with respect to the angle B,
and CC' and CK with respect to C. Therefore the
three lines AA', BB', CC', are concurrent: their point
of concurrence is the isotomic conjugate of K with respect to the triangle ABC.
CoR. 1.-The Brocardpoints are on the Brocard circle.
COR. 2.-The sides of the triangle FDE are parallel to
the lines AQ, BO, CD, respectively.