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 ISOGONAL AND ISOTOMIC POINTS. 169

DEF.-Two points, X, Y, are said to be isotomic conjugates, with respect to a triangle ABC, when the pairs of
lines AX, AY; BX, BY; CX, CY are isotomic conjugates, with respect to the sides BC, CA, AB respectively.
Exercises.
1. If the multiples for which the point X is the mean centre of
I 1
the points A, B, C be a, 3, y; prove that-,, are the mula ) 7
tiples for which the isotomic conjugate of X is the mean centre of
the points A, B, C.
2. If a right line meet the sides of a triangle in the points A',
B', C'; prove that the triads of points in which the isogonal and
the isotomic conjugates of AA', BB', CC', with respect to the
angles A, B, C, meet the sides of the triangle, are each collinear.
DEF.-Lines BC, B'C' are said to be antiparallel, with
A
B C
respect to the angle A, when the angle ABC is equal to
AC'B'.
There are three systems of antiparallels with respect to a triangle ABC.
1~.-Antiparallels to BC with respect to the opposite
angle A.
2~,, CA,, B.
30., AB,, C.
Prop. 4.-The antiparallels to the sides of a triangle
are parallel to the tangents to its circumcircle at the
angular points.