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. 151
31. If a variable line meet four fixed lines in points whose
anharmonic ratio is constant, it cuts these four lines homographically.
32. Given the I CD to the diameter AB of a semicircle, it is
required to draw through A a chord, cutting CD in E and the
semicircle in F, such that the ratio of CE: EF may be given.
33. Draw in the last construction the line AEF so that the
quadrilateral CEFB may be a maximum.
34. The O described through the centres of the three escribed
Os of a plane A, and the circumscribed ( of the same A, will
have the centre of the inscribed (D of the A for one of their centres
of similitude.
35. The Os on the diagonals of a complete quadrilateral cut
orthogonally the 0 described about the A formed by the three
diagonals.
36. When the three Is from the vertices of one A on the
sides of another are concurrent, the three corresponding Is from
the vertices of the latter, on the sides of the former, are concurrent.
37. If a 0 be inscribed in a quadrant of a 0; and a second 0
be described touching the O, the quadrant, and radius of quadrant;
and a _ be let fall from the centre of the second O on the line
passing through the centres of the first 0 and of the quadrant;
then the A whose angular points are the foot of the I, the
centre of the quadrant, and the centre of the second 0, has its
sides in arithmetical progression.
38. In the last Proposition, the Is let fall from the centre of
the second 0 on the radii of the quadrants are in the ratio of
1: 7.
39. When three Os of a coaxal system touch the three sides of
a A at three points, which are either collinear or concurrently
connectant with the opposite vertices, their three centres form.,
with those of the three Os of the system which pass through the
vertices of the A, a system of six points in involution.
40. If two Os be so placed that a quadrilateral may be inscribed in one and circumscribed to the other, the diagonals of
the quadrilateral intersect in one of the limiting points,
41. If from a fixed point Is be let fall on two conjugate rays
of a pencil in involution, the feet of the I s are collinear with a
fixed point.
42. MIQUEL's THEOREM.-If the five sides of any pentagon
ABCDE be produced, forming five As external to the pentagon,