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.

MISCELLANEOUS EXERCISES. 241
5~. When the sum of the squares of the sides of the triangle
DiD2D3, each multiplied by a given constant, is given.
6~. When the Brocard angle of the triangle D1D2D3 is given.
73. The poles of the sides of the triangle ABC, with respect to
the corresponding M'Cay's circles, are the vertices of Brocard's
first triangle.
74. The mean centre of three corresponding points in the
system of figures, Ex. 60, for the system of multiplies a2, b2, c2,
is the symmedian point of the triangle ABC.
75. If from the middle points of the sides of the triangle ABC
tangents be drawn to the corresponding Neuberg's circles, the
points of contact lie on two right lines through the centroid of
ABC.
76. The circumcentre of a triangle, its symmedian point, and
the orthocentre of its pedal triangle, are collinear. (TUCxER.)
77. The orthocentre of a triangle, its symmedian point, and
the orthocentre of its pedal triangle, are collinear. (E. VAN
AUBEL.)
78. The perpendicular from the angular points of the triangle
ABC on the sides of Brocard's first triangle are concurrent, and
their point of concurrence (called TARRY'S POINT) is on the
circumcircle of ABC.
79. The Simson's line of Tarry's point is perpendicular to OK.
80. The parallels drawn through A, B, C to the sides (B'C'
C'A', A'B'), or to (C'A', A'B', B'C'), or (A'B', B'C', C'A') of
the first triangle of Brocard, concur in three points, R, R', R".
(NEUBERG.)
81. The triangles RR'R", ABC have the same centroid. (Ibid.)
82. R is the point on the circumcircle whose Simson's line is
parallel to OK.
83. If from Tarry's point, Ex. 78, perpendiculars be drawn to
the sides BC, CA, AB of the triangle, meeting those sides in (a
ai, a2), (i3, 1, 2), (7Y, 7l, 72), the points a, 31, y, are collinear
(Simson's line). So also al,.82, 7, and a2, $, '7 are collinear
systems. (NEUBERG.)
84. M'Cay's circles are the inverses of the sides of Brocard's
first triangle, with respect to the circle whose centre is the centroid of ABC, and which cuts its Brocard circle orthogonally.