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.

BOOK VI. 157
76. Given three rays A, B, C, find three other rays X, Y, Z
through the same vertex 0, so that the anharmonic ratios of the
pencils (. ABXY), (0. BCYZ), (0. CAZX), may be given.
77. If a A similar to that formed by the centres of three given
Os slide with its three vertices on their circumferences, the vertices divide the Os homographically.
78. Find the locus of the centre of a (, being given that the
polar of a given point A passes through a given point B, and
the polar of another given point C passes through a given
point D.
79. If a A be self-conjugate with respect to a given 0, the
0 described about the A is orthogonal to another given circle.
80. The Os self-conjugate to the As formed by four lines are
coaxal.
81. The pencil formed by lines II to the sides and diagonals of a
quadrilateral is involution.
82. If four Os be co-orthogonal, that is, have a common orthogonal 0, their radical axes form a pencil in involution.
83. In a given ( to inscribe a A whose sides shall divide in a
given anharmonic ratio given arcs of the circle.
84. When four Os have a common point of intersection, their
six radical axes form a pencil in involution.
85. The pencil formed by drawing tangents from any point in
their radical axis to two 0 s, and drawing two lines to their centres
of similitude, is in involution.
*86. If a pair of the opposite L s of a quadrilateral be equal to a
right L, then the sum of the squares of the rectangles contained
by the opposite sides is equal to the square of the rectangle contained by the diagonals.
87. Prove that the problem 17, page 38, " To inscribe in a given
A DEF, a A given in species whose area shall be a minimum,"
admits of two solutions; and also that the point O' in the second
solution, which corresponds to 0 in the first, is the inverse of 0
with respect to the circle which circumscribes the A DEF.
88. The line joining the intersection of the Ls of a A to the
centre of a circumscribed 0 is I to the axis of perspective of the
given A, and the A formed by joining the feet of the I s.

* This Theorem is due to Bellavitis. See his Aethode des Equziollences.