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.

164 A SEQUEL TO EUCLID.
135. If R, r be the radii of two circles, C, C', of which the
former is supposed to include the latter; then if a series of circles
01, 02, 03,...m be described touching both and touching each
other in succession, prove that if traversing the space between
C, C' n times consecutively the circle Om touch O1 if 8 be the
distance between the centres,
(R - r)2 - 4Rr tan n = 58. -(STEINER.)
136. If A, B, C, D be any four points, and if the three pairs
of lines which join them intersect in the points b, c, d, then the
nine-points circles of the four triangles ABC, ABD, ACD, BCD,
and the circle about the triangle bed, all pass through a common
point P.
137. If Ai, B1, C1, Di be the orthocentres of the triangles
A, B, C, &c., and b1, cl, di the points determined by joining
A1, B1, C1, Di, in pairs; then the nine-points circles of the four
triangles A1, B1, C1, &c., and the circumscribed circle of the triangle blcldl, all pass through the former point P.-(Ex. 29.)
138. The SIMSON'S lines (Book II. Prop. 12) of the extremities of any diameter of the circumcircle of a triangle intersect at
right angles on the nine-points circle of the triangle.
139.* Every tangent to a circle is cut harmonically by the sides
of a circumscribed square, and also by the sides of a circumscribed
trapezoid whose non-parallel sides are equal.
140. A variable chord of a circle passes through a fixed point;
its extremities and the fixed point are joined to the centre; prove
that the circumcircles of the three triangles so formed touch in
every position a pair of circles belonging to two given coaxal
systems.
141. Weill's Tieorem.-If two circles be so related that a polygon of n sides can be inscribed in one and circumscribed to the
other, the mean centre of the points of contact is a fixed point.
142. In the same case the locus of the mean centre of any
number (n - r) of the points is a circle.
Weill's Theorem was published in Liouvile's Journal,
Third Series, Tome IV., page 270, for the year 1878. A
proposition, of which Weill's is an immediate inference, was
published by the Author in 1862, in the Quarterly Journal
of Pure and Applied Mathematics, Vol. V., page 44, Cor. 2.
* Theorems 139, 140, have been communicated to the author
by ROBERT GRAHAM, Esq., A.M., T.C.D.