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.

242 A SEQUEL TO EUCLID.
85. The tangents from ABC to the Brocard circle are proportional to a-1, b-1, c-1.
86. If the alternate sides of Lemoine's hexagon be produced to
meet, forming a second triangle, its inscribed circle is equal to
the nine-points circle of the original triangle.
87. If K be the symmedian point of the triangle ABC, and
the angles ABK, BCK, CAK be denoted by e1, 0e, 03, respectively; and the angles BAK, CBK, ACK by (1, P2, P3,
respectively; then cot 01 + cot 02 + cot 03 = cot pi + cot IP2
+ cot p3 = 3 cot ow. (TCKEIR.)
88. If Al, B1, C1 be the vertices of Brocard's first triangle, the
lines BA1, AB1 are divided proportionally by nr'.
89. The middle point of AB, A1Bi, nn' are collinear (STOLL.)
90. The triangle formed by the middle points of Al, B1, C1,
is in perspective with ABC. (Ibid.)
91. If the Brocard circle of ABC intersect BC in the points
M, M', the lines AM, AM' are isogonal conjugates with respect
to the angle BAC.
92. If n, r' be the Brocard points of a harmonic polygon of
asides, Q a' = 2R sin w (cos2o - sin2w. tan2 _).
93. If the polars of the points B, C, with respect to the Brocard
circle of the triangle ABC, intersect the side BC in the points
L, L', respectively; the lines AL, AL' are isogonal conjugates
with respect to the angle BAC.
94. The reciprocal of any triangle with respect to a circle,
whose centre is either of the Brocard points, is a similar triangle,
having the centre of reciprocation for one of its Brocard points.
95. If the angles which the sides AB, BC, CD... KL of a
harmonic polygon subtend at any point of its circumcircle be
denoted by a, B, 7,... A, the perpendiculars from the Brocard
points on the sides are proportional respectively to the quantities,
sin x cosec A, sin a cosec B, sin 6 cosec C,... sin K cosec L,
and their reciprocals.
96. The triangle ABC, its reciprocal with respect to the Brocard circle, and the triangle pqr [Section III., Prop. 2, Cor. 2],
are, two by two, in perspective, and have a common axis of perspective.