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.

ADDITIONAL PROPOSITIONS AND EXERCISES. 163
125. If r be the radius of the inscribed circle of a triangle ABC,
and p the radius of a circle touching the circumscribed circle internally and the sides AB, AC; then p cos2 'A = r.
126. Prove the corresponding relation p' cos2 'A = r' for the
ease of external contact.
127. Prove by inversion the equality of the two circles in
Prop. 8, Cor. 4, p. 32.
128. If AB, CD be the diameters of two circles, and be also
segments of the same line, prove that the two circles are equal
which touch respectively the circles on AB, CD; their radical
axis on opposite sides, and any circle whose centre is the middle
point of AD.-(STEINER.)
129. Given three points, A, B, C, and three multiples, I, m, n,
find a point 0 such that IAO + mBO + nCO may be a minimum.
130. If A, B, C, D be any four points connected by four circles,
each passing through three of the points, then not only is the
angle at A between the arcs ABC, ADC equal to the angle at C
between CDA, CBA, but it is also equal to the angle at D between the arcs DAB, DCB; and to the angle at B between BCD,
BAD.-(HAMILTON.)
131. If A, B, C be the escribed circles of a triangle, and if
A', B', C' be three other circles touching ABC as follows, viz.
each of them touching two of the former exteriorly, and one interiorly; then A', B', C' intersect in a common point P, and the
lines of connexion at P with the centres of the circles are perpendicular to the sides of the triangle.
132. The line of collinearity of the middle points of the diagonals of a complete quadrilateral is perpendicular to the line of
collinearity of the orthocentres of the four triangles.
133. The sines of the angles which the line of collinearity of
the middle points of the diagonals of a complete quadrilateral
makes with the sides are proportional to the diameters of the
circles described about its four triangles.
134. If r, p be the radii of two concentric circles, and R the
radius of a third circle (not necessarily concentric), so related to
them that a triangle described about the circle r may be inscribed
in R, and a quadrilateral about p may be inscribed in R: then
rip + p/R = p/r.