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.

153

44. Through the vertices of any A are drawn any three parallel lines, and through each vertex a line is drawn, making the
same L with one of the adjacent sides which the parallel makes
with the other; these three lines are concurrent. Required the
locus of the point in which they meet.
45. If from any point in a given line two tangents be drawn to
a given (, X, and if a (, Y, be described touching X and the two
tangents, the envelope of the polar of the centre of Y with respect
to X is a circle.
46. The extremities of a variable chord XY of a given ( are
joined to the extremities of a fixed chord AB; then, if m AX. AY
+ n BX. BY be given, the envelope of XY is a circle.
47. If A, A' be conjugate points of a system in involution, and
if AQ, A'Q be I to the lines joining A, A' to any fixed point P, it
is required to find the locus of Q.
48. If a, a', b, b',, ', be three pairs of conjugate points of a
system in involution; then,
(1). ab'. b'. ca' = - a'. b'c. o'a.
(2). ab'. be. c = - a'b. b'c'. ca.
ab. a' a'b. a'b'
(3).
a(. ac' c a'e. c'
49. Construct a right-angled A, being given the sum of the
base and hypotenuse, and the sum of the base and perpendicular.
50. Given the perimeter of a right-angled A whose sides are
in arithmetical progression: construct it.
51. Given a point in the side of a A; inscribe in it another A
similar to a given A, and having one L at the given point.
52. Given a point D in the base AB produced of a given A ABC;
draw a line EF through D cutting the sides so that the area of
the A EFC may be given.
53. Construct a A whose three L s shall be on given Os,
and whose sides shall pass through three of their centres of
similitude.
54. From a given point 0 three lines OA, OB, OC are drawn
to a given line ABC; prove that if the radii of the Os inscribed
in OAB, OBC are given, the radius of the ( inscribed in OAC
will be determined.
55. Equal portions OA, OB are taken on the sides of a given
right L AOB, the point A is joined to a fixed point C, and a I let
fall on AC from B: the locus of the foot of this 1 is a circle.