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.

156 A SEQUEL TO EUCLID.
63. The sum of the squares of the radii of the four Os, cutting orthogonally the inscribed and escribed Os of a plane A,
taken three by three, is equal to the square of the diameter of the
circumscribed 0.
64. Describe through two given points a ( cutting a given arc
of a given 0 in a given anharmonic ratio.
65. All Os which cut three fixed Os at equal L s form a coaxal
system.
66. Being given five points and a line, find a point on the line,
so that the pencil formed by joining it to the five given points
shall form an involution with the line itself.
67. If a quadrilateral be inscribed in a circle, the circle described on the third diagonal as diameter will be the circle of
similitude of the circles described on the other diagonals as diameters.
68. If ABC be any A, B'C' a line drawn 11 to the base BC;
then, if O, O' be the escribed Os to ABC, opposite the L s B and
C respectively, 01 the inscribed 0 of AB'C', and O'1 the escribed
O opposite the L A; then, besides the lines AB, AC, which are
common tangents, 0, O', 01, O'1, are all touched by two other
circles.
69. When two Os intersect orthogonally, the locus of the point
whence four tangents can be drawn to the Os, and forming a
harmonic pencil, consists of two lines, viz., the polars of the
centre of similitude of the two circles.
70. If two lines be divided homographically in the two systems of points a, b, c, &c., a', b', c', &c., then the locus of the
points of intersection of ab', a'b, ac', a'c, ad', a'd, &c., is a right
line.
71. Being given two homographic pencils, if through the point
of intersection of two corresponding rays we draw two transversals, which meet the two pencils in two series of points, the lines
joining corresponding points of intersection are concurrent.
72. Inscribe a A in a ( having two sides passing through two
given points, and the third 11 to a given line.
73. If two As be described about a 0, the six angular points
are such that any four are subtended equianharmonically by the
other two.
74. Given four points A, B, C, D on a given line, find two
other points X, Y, so that the anharmonicratios (ABXY), (CDXY)
may be given.
75. If two quadrilaterals have the same diagonals, the eight
points of intersection of their sides are such that any four are
subtended equianharmonically by the other four.