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SECTION V. Page 153.
1. A STRAIGHT line of given length being drawn from the
centre at right angles to the plane of a circle; to determine that
-point in it which is equally distant from the upper end of the line,
and the circumference of the circle.
2. To determine a point in a line given in position, to which
lines drawn from two given points may have the greatest difference
possible.
3. A straight line being divided in two given points; to deter-
mine a third such that its distances from the extremities may be
proportional to its distances from the given points.
4. In a straight line given in position, to determine a point, at
which two straight lines drawn from given points on the same side,
will contain the greatest angle.
5. To determine the position of a point, at which lines drawn
from three given points shall make with each other angles equal to
given angles.
6. To divide a straight line into two parts such that the rectangle
contained by them may be equal to the square of their difference.
7. If a straight line be divided into any two parts; to produce it
so that the rectangle contained by the whole line so produced and the
part produced, may be equal to the rectangle contained by the given
line and one segment.
COR. 1. To produce the line, so that the rectangle contained by
the whole line and the part produced may be equal to the rectangle
contained by two given lines.
COR. 2. To produce the line, so that the rectangle contained by
the whole line produced and the part produced may be equal to a
given square.
8. To determine two lines such that the sum of their squares
may be equal to a given square, and their rectangle equal to a given
rectangle.
9. To divide a straight line into two parts, so that the rectangle
contained by the whole and one of the parts may be equal to the
square of a given line, which is less than the line to be divided.
10. To divide a given line into two such parts, that the rectangle
contained by the whole line, and one of the parts may be (m) times
the square of the other part; (m) being whole or fractional.
1.1. To divide a given line into two such parts, that the square
of the one shall be equal to the rectangle contained by the other and
a given line.
12. A straight line being given in magnitude and position; to
draw to it, from a given point, two lines, whose rectangle shall be
equal to a given rectangle, and which shall cut off equal segments
from the given line.
13. To draw a straight line which shall touch a given circle, and
make with a given line, an angle equal to a given angle.
14. Through a given point to draw a line terminating in two
lines given in position, so that the rectangle contained by the two
parts may be equal to a given rectangle.
15. From a given point to draw a line cutting two given parallel
lines, so that the difference of its segments may be equal to a given
line.
16. From a given point without a circle, to draw a straight line
cutting the circle, so that the rectangle contained by the part of it
without, and the part within the circle shall be equal to a given
square.
17. From a given point in the circumference of a semicircle, to
draw a straight line meeting the diameter, so that the difference
between the squares of this line and a perpendicular to the diameter
from the point of intersection may be equal to a given rectangle.
18. From a given point to draw two lines to a third given in
position, so that the rectangle contained by those lines may be equal
to a given rectangle, and the difference of the angles which they
make with that part of the third which is intercepted between them
may be equal to a given angle.
19. Two points being given without a given circle; to determine
a point in the circumference, from which lines drawn to the two
given points shall contain the greatest possible angle.
20. From the bisection of a given arc of a circle, to draw a
straight line such that the part of it intercepted between the chord of
that and the opposite circumference shall be equal to a given straight
21. To draw a straight line through a given point, so that the
sum of the perpendiculars to it from two other given points may be
equal to a given line.
22. To draw a straight line through one of three points given in
position, so that the rectangle contained by the perpendiculars let
fall upon it from the other two, may be equal to a given square.
23. A given straight line being divided into two parts; to cut off
a part which shall be a mean proportional between the two remaining
segments.
24. To draw a straight line making a given angle with one of the
sides of a given triangle, so that the triangle cut off may be to the
whole in a given ratio.
25. Between two given straight lines containing a given angle, to
place a straight line of given length, and subtending that angle, so that
the segment of the one of them adjacent to the angle may be to the
segment of the other which is not adjacent, in the ratio of two given
lines.
26. From two given points to draw two lines to a point in a third,
such that the difference of their squares may be equal to a given
27. To divide a given straight line into two such parts, that the
square of one may be to the excess of a given rectangle above the
square of the other, in a given ratio.
28. From any angle of a triangle, not isosceles about the angle,
to draw a line without the triangle to the opposite side produced,
which shall be a mean proportional between the segments of the side.
29. From the obtuse angle of any triangle, to draw a line within
the triangle to the opposite side, which shall be a mean proportional
between the segments of the side.
30. From the common extremity of the diameters of two semi-
circles, given in magnitude and position; to draw a line meeting the
circumferences so that the rectangle contained by the two chords
may be equal to a given square.
31. To draw a line parallel to a given line, which shall be termi-
nated by two others given in position, so as to form with them a
triangle equal to a given rectilineal figure.
32. To bisect a triangle by a line drawn parallel to one of its
sides.
33. To divide a given triangle into any number of parts, having
a given ratio to each other, by lines drawn parallel to one of the
sides of the triangle.
34. To divide a given triangle into any number of equal parts,
by lines drawn parallel to a given line.
35. To divide a trapezium, which has two sides parallel, into
any number of equal parts, by lines drawn parallel to those sides.
36. From one of the angular points of a given square, to draw
a line meeting one of the opposite sides and the other produced, in
such a manner, that the exterior triangle formed thereby may have
a given ratio to the square.
37. From a given point in the side produced of a given rectangu-
lar parallelogram, to draw a line which shall cut the perpendicular
sides and the other side produced, so that the trapezium cut off,
which stands on the aforesaid side, may be to the triangle which
stands upon the produced part of the opposite side, in a given ratio.
38. Through a given point between two straight lines containing
a given angle, to draw a line which shall cut off a triangle equal to
a given figure.
39. Between two lines given in position, to draw a line equal to
a given line, so that the triangle thus formed may be equal to a
given rectilineal figure.
40. From two given lines to cut off two others, so that the re-
mainder of one may have to the part cut off from the other, a given
ratio; and the difference of the squares of the other remainder and
part cut off from the first may be equal to a given square.
41. From two given lines to cut off two others which shall have
a given ratio, so that the difference of the squares of the remainders
42. From two given lines to cut off two others, so that the re-
mainders may have a given ratio, and the sum of the squares of the
parts cut off may be equal to the square of a given line.
43. Two points being given in a given straight line; to determine
a third, such that the rectangles contained by its distances from each
extremity and the given point adjacent to that extremity may be
equal.
44. Through the point of intersection of two given circles, to
draw a line in such a manner, that the sum of the respective
rectangles contained by the parts thereof, which are intercepted
between the said point and their circumferences, and given lines
A and B, may be equal to a given square.
45. Through a given point, to draw an indefinite line such, that
if lines be drawn from two other given points and forming given
angles with it, the rectangle contained by the segments intercepted
between the given point and the two lines so drawn, shall be equal to
the square of a given line.
46. Through a given point between two straight lines containing
a given angle, to draw a line such that a perpendicular upon it from
the given angle may have a given ratio to a line drawn from one ex-
tremity of it, parallel to a line given in position.
47. Through a given point between two indefinite straight lines,
not parallel to one another, to draw a line, which shall be terminated
by them, so that the rectangle contained by its segments shall be less
than the rectangle contained by the segments of any other line drawn
through the same point.
SECTION VI. Page 184.
1. To describe an isosceles triangle on a given finite straight
2. To describe a square which shall be equal to the difference
of two squares, whose sides are given.
COR. Hence a mean proportional between the sum and difference
of two given lines may be determined.