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.

245

Brocard circle form the vertices of a harmonic polygon, and the
Brocard circle of the latter polygon is coaxal with the Brocard
circle and circumcircle of the former.
119. If the symmedian lines AK, BK, CK of a triangle meet
the sides BC, CA, AB in the points Ka, Kb, Kc, respectively, the
triangle ABC, its reciprocal with respect to its circumcircle, and
the triangle KaKbKc, have a common axis of perspective.
120. If DE be the diameter of the circumcircle of ABC, which
bisects BC, and KaE be the intersection of the symmedian AK
with BC, the line joining the middle point of BC to K meets the
lines EKa, DIKa, respectively, on the bisectors, internal and external, of the angle ABC.
121. If a heptagon circumscribed to a circle has for points of
contact the vertices of a harmonic heptagon, the seven hexagons
obtained by omitting in succession a side of the original poylgon
have a common Brianchon point.
122. Prove ~ sin A cos (A + w) = 0.
123. If 0 be the circumcentre of a harmonic polygon of n
sides, L one of the limiting points of the circumcircle and
Brocard circle; then if OL = R cot 0,
cos 20 = tan co tan-.
n
124. If ABCD, &c., be a harmonic polygon, and if a circle
described through the pairs of points A, B; B, C; C, D, &c.,
touch the radical axis of the circumcircle and Brocard circle, the
points of contact are in involution.
125. Any four given sides of a harmonic polygon meet any of
the remaining sides in four points, whose anharmonic ratio is
constant.
126. The quadrilateral formed by any four sides of a harmonic
polygon is such, that the angles subtended at either of the
Brocard points, by opposite sides, are supplemental.
127. The reciprocals of two co-symmedian triangles, with respect to their common symmedian point, are equiangular.
128. The reciprocals of two co-symmedian triangles, with respect to either of their common Brocard points, are two cosymmedian triangles.
129. The reciprocal of a harmonic polygon, with respect to
either of its Brocard points, is a harmonic polygon, having the
centre of reciprocation for one of its Brocard points.
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