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.

149

9. The I from the right angle on the hypotenuse of a rightangled A is a harmonic mean between the segments of the hypotenuse made by the point of contact of the inscribed circle.
10. If a line be cut harmonically by two Os, the locus of the
foot of the I, let fall on it from either centre, is a 0, and it cuts
any two positions of itself homographically (see Prop. 3, Cor. 2,
Section VII.).
11. Through a given point to draw a line, cutting the sides of
a given A in three points, such that the anharmonic ratio of the
system, consisting of the given point and the points of section,
may be given.
12. If squares be described on the sides of a A and their centres joined, the area of the A so formed exceeds the area of the
given triangle by -th part of the sum of the squares.
13. The locus of the centre of a 0 bisecting the circumferences
of two fixed Os is a right line.
14. Divide a given semicircle into two parts by a I to the
diameter, so that the diameters of the Os described in them may
be in a given ratio.
15. The side of the square inscribed in a A is half the harmonic mean between the base and perpendicular.
16. The Os described on the three diagonals of a quadrilatera
are coaxal.
17. If X, X' be the points where the bisectors of the L A of
a A and of its supplement meet the side BC, and if Y, Y'; Z, Z',
be points similarly determined on the sides CA, AB; then
1 1 1 J
xx+ Y Y ZZ'
a2 b2 c2
and 3+ + =0.
anxx' + YY' ZZ'
18. Prove Ptolemy's Theorem, and its converse, by inversion
19. A line of given length slides between two fixed lines: find
the locus of the intersection of the Is to the fixed lines from the
extremities of the sliding line, and of the Is on the fixed lines
from the extremities of the sliding line.
20. If from a variable point P Is be drawn to three sides of
a A; then, if the area of the A formed by joining the feet of
these Is be given, the locus of P is a circle.