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 IV.

65

16. If, 0, 0', 0",, be the centres of the inscribed and
escribed Os of a plane A, then 0 is the mean centre of the points
0', 0", 0"', for the system of multiples (s - a), (s - b), (s - c).
17. In the same case, 0' is the mean centre of the points
0, 0", 0"', for the system of multiples s, s - b, s - c, and corresponding properties hold for the points 0", 0"'.
18. If r be the radius of the ( inscribed in a A, and pl, p2 the
radii of two Os touching the circumscribed 0, and also touching
each other at the centre of the inscribed D0; then
2 1 1
r pi p2
19. If r, ri, r2, r3 be the radii of the inscribed and escribed Os
of a plane A, and R the radius of the circumscribed 0; then
rl + r2 + r3 - r = 4R.
20. vn the same case,
1 1 1 1
-+= -+ - —.
r '1 r2 r'3
21. In a given 0 inscribe a A, so that two of its sides may
pass through given points, and that the third side may be a
maximum.
22. What theorem analogous to 18 holds for escribed (Os
23. Draw from the vertical L of an obtuse-angled A a line
to a point in the base, such that its square will be equal to the
rectangle contained by the segments of the base.
24. If the line AD, bisecting the vertical L A of the A ABC,
meets the base BC in D, and the circumscribed ( in E, then the
line CE is a tangent to the ( described about the A ADC.
25. The sum of the squares of the I s from the angular points
of a regular polygon inscribed in a 0 upon any diameter of the
( is equal to half n times the square of the radius.
26. Given the base and vertical L of a A, find the locus of
the centre of the ( which passes through the centres of the three
escribed circles.
27. If a 0 touch the arcs AC, BC, and the line AB in the
construction of Euclid (I. i.), prove its radius equal to 8 of AB.
28. Given the base and the vertical L of a A, find the locus
of the centre of its " Nine-point Circle."
F