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.

244 A SEQUEL TO EUCLID.
107. Prove a corresponding property for isotomic conjugates.
108. In fig., p. 207, prove that
OS: OS':: cos (C -):cos ( +
109. If AA', BB', CC' be fixed chords of a circle X, and
circles cutting X at equal angles be described through the points
A, A'; B, B'; C, C', respectively, the locus of their radical
centre is a right line. (M'CAY.)
110. Find the locus of a point in the plane of a triangle, which
is such that the triangle formed by joining the feet of its perpendiculars may have a given Brocard angle.
111. If the extremities of the base of a triangle be given in
position, and also the symmedian passing through one of the
extremities, the locus of the vertex is a circle.
112. If through the extremities A, B; B, C; C, D, &c., of
the sides of a harmonic polygon circles be described touching the
Brocard circle, the contacts being all of the same species, these
circles cut the circumcircle at equal angles, and are all tangential to a circle coaxal with the Brocard circle and circumcircle.
113. The radical axis of the circumcircle and cosine circles of
a harmonic quadrilateral passes through the symmedian point of
the quadrilateral.
114. If the Brocard angles of two harmonic polygons, A, B, be
complementary, and if the cosine circle ofA be the circumcircle
of B, the cosine circle of B is equal to the circumcircle of A.
115-117. In the same case, if the cosine circle of B coincide
with the circumcircle of A; and n, i' be the numbers of sides of
the polygons; w, w' their Brocard angles; 8 the diameter of
their common Brocard circle; then7r 7r
10. tan c = cos - C- cos -.
X n
2~. The Brocard points of A coincide with those of B.
3. 2 COS2 =R2 COS
118. If through the vertices of a harmonic polygon of any
number of sides circles be described, cutting its circumcircle and
Brocard circle orthogonally, their points of intersection with the