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.

158 A SEQUEL TO EUCLID.
89. If two Os whose radii are R, R', and the distance of whose
centres is 8, be such that a hexagon can be inscribed in one and
circumscribed to the other; then
1 1
(R - Y)2 + 4R'2R6 + (R2- 62)2 - 4R2
1
21/2 (R2 + 82) - (R2 - 53)2'
90. In the same case, if an octagon be inscribed in one and
circumscribed to the other,
= (R2 - 2)2 + + R
2R'2 (R2 + 82) - (Ri 8)2 1)
91. If a variable 0 touch two fixed Os, the polar of its centre
with respect to either of the fixed Os touches a fixed circle.
92. If a 0 touch three Os, the polar of its centre, with respect
to any of the three Os, is a common tangent to two circles.
*93. Prove that the Problem, to inscribe a quadrilateral, whose
perimeter is a minimum in another quadrilateral, is indeterminate
or impossible, according as the given quadrilateral has the sum of
its opposite angles equal or not equal to two right angles.
94. If a quadrilateral be inscribed in a 0, the lines joining the
feet of the I s, let fall on its sides from the point of intersection
of its diagonals, will form an inscribed quadrilateral Q of minimum
perimeter; and an indefinite number of other quadrilaterals may be
inscribed whose sides are respectively equal to the sides of Q, the
perimeter of each of them being equal to the perimeter of Q.
95. The perimeter of Q is equal to the rectangle contained by
the diagonals of the original quadrilateral divided by the radius of
the circumscribed circle.
96. Being given four lines forming four As, the sixteen centres of the inscribed and escribed Os to these A s lie four by four
on four coaxal circles.
97. If the base of a A be given, both in magnitude and position, and the ratio of the sum of the squares of the sides to the
area, the locus of the vertex is a circle.

* The Theorems 87 and 93-96 have been communicated to the author
by Mr. W. S. M'CAY, F.T.C.D.