33. If from any point within a rectangular parallelogram lines be

drawn to the angular points; the sums of the squares of those which

are drawn to the opposite angles are equal.

34. The squares of the diagonals of a parallelogram are together

equal to the squares of the four sides.

35. If two sides of a trapezium be parallel to each other; the

squares of its diagonals are together equal to the squares of its two

sides which are not parallel, and twice the rectangle contained by

its parallel sides.

36. The squares of the diagonals of a trapezium are together

double the squares of the two lines joining the bisections of the op-

posite sides.

37. The squares of the diagonals of a trapezium are together

less than the squares of the four sides, by four times the square of

the line joining the points of bisection of the diagonals.

38. In any trapezium, if two opposite sides be bisected; the

sum of the squares of the two other sides, together with the squares

of the diagonals, is equal to the sum of the squares of the bisected

sides, together with four times the square of the line joining those

points of bisection.

39. If squares be described on the sides of a right-angled

triangle; each of the lines joining the acute angles and the opposite

angle of the square, will cut off from the triangle an obtuse-angled

triangle, which will be equal to that cut off from the square by a

line drawn from the intersection with the side to that angle of the

square which is opposite to it.

40. If squares be described on the two sides of a right-angled

triangle; the lines joining each of the acute angles of the triangle

and the opposite angle of the square will meet the perpendicular

drawn from the right angle upon the hypothenuse, in the same point.

41. If squares be described on the three sides of a right-angled

triangle, and the extremities of the adjacent sides be joined; the

triangles so formed are equal to the given triangle and to each other.

42. If the sides of the square described on the hypothenuse of

a right-angled triangle be produced to meet the sides (produced if

d

necessary) of the squares described upon the legs; they will cut off
triangles equiangular and equal to the given triangle.

43. If from the angular points of the squares described upon the

sides of a right-angled triangle perpendiculars be let fall upon the

hypothenuse produced; they will cut off equal segments; and the

perpendiculars will together be equal to the hypothenuse.

44. If on the two sides of a right-angled triangle squares be

described; the lines joining the acute angles of the triangle and the

opposite angles of the squares will cut off equal segments from the

sides and each of these equal segments will be a mean proportional

between the remaining segments.

45. If squares be described on the hypothenuse and sides of a

right-angled triangle, and the extremities of the sides of the former

and the adjacent sides of the others be joined; the sum of the

squares of the lines joining them will be equal to five times the

square of the hypothenuse.

46. If a line be drawn parallel to the base of a triangle, and

terminated in the sides; to draw a line cutting it, and terminated

also by the sides, so that the rectangle contained by their segments

may be equal.

47. If the sides, or sides produced, of a triangle be cut by any

line; the solids formed by the segments which have not a common

extremity are equal.

48. If through any point within a triangle, three lines be drawn

parallel to the sides; the solids formed by the alternate segments of

these lines are equal.

49. If through any point within a triangle lines be drawn from

the angles to cut the opposite sides; the segments of any one side

will be to each other in the ratio compounded of the ratios of the

segments of the other sides.

50. If from each of the angles of any triangle, a line be drawn

through any point within the triangle to the opposite side; the solid

formed by the segments thereof, intercepted between the angles and

the point, will have to the solid formed by the three remaining

segments, the same ratio that the solid formed by the three sides

of the triangle has to either of the (equal) solids formed by the

alternate segments of the sides.

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.

11. 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

line.

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

square.

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