SECTION IV. Page 120.

1. THE diameters of a rhombus bisect each other at right

angles.

2. If the opposite sides or opposite angles of a quadrilateral

figure be equal; the figure will be a parallelogram.

3. To bisect a parallelogram by a line drawn from a point in one

of its sides.

4. If from any point in the diameter (or diameter produced) of

a parallelogram straight lines be drawn to the opposite angles; they

will cut off equal triangles.

5. From one of the angles of a parallelogram to draw a line to

the opposite side, which shall be equal to that side together with the

segment of it which is intercepted between the line and the opposite

angle.

6. If from one of the angles of a parallelogram a straight line be

drawn, cutting the diameter, a side and a side produced; the seg-

ment intercepted between the angle and the diameter is a mean

proportional between the segments intercepted between the diameter

and the sides.

7. The two triangles, formed by drawing straight lines from any

point within a parallelogram to the extremities of two opposite sides,

are together half of the parallelogram.

8. If a straight line be drawn parallel to one of the sides of a

parallelogram, and one extremity of this line be joined to the opposite

one of the parallel side, by a line which also cuts the diameter; the

segments of the diameter made by this line will be reciprocally pro-

portional to the segments of that part of it which is intercepted

between the side and the parallel line.

9. If two lines be drawn parallel and equal to the adjacent sides

of a parallelogram; the lines joining their extremities, if produced,

will meet the diameter in the same point.

10. If in the sides of a square, at equal distances from the four

angles, four other points be taken, one in each side; the figure con-

tained by the straight lines which join them shall also be a square.

11. The sum of the diagonals of a trapezium is less than the sum

of any
four lines which can be drawn to the four angles from any

point within the figure, except from the intersection of the diagonals.

12. Every trapezium is divided by its diagonals into four tri-

angles proportional to each other.

13. If two opposite angles of a trapezium be right angles; the

angles subtended by either side at the two opposite angular points

shall be equal.

14. To determine the figure formed by joining the points of

bisection of the sides of a trapezium; and its ratio to the trapezium.

15. To determine the figure formed by joining the points where

the diagonals of the trapezium cut the parallelogram (in the last

problem); and its ratio to the trapezium.

16. If two sides of a trapezium be parallel; its area is equal to

half that of a parallelogram whose base is the sum of those two sides,

and altitude the perpendicular distance between them.

17. If from any angle of a rectangular parallelogram a line be

drawn to the opposite side, and from the adjacent angle of the tra-

pezium thus formed another be drawn perpendicular to the former;

the rectangle contained by these two lines is equal to the given

parallelogram.

18. To divide a parallelogram into two parts which shall have a

given ratio, by a line drawn parallel to a given line.

19. To bisect a trapezium by a line drawn from one of its angles.

20. To bisect a trapezium by a line drawn from a given point in

one of its sides.

21. If two sides of a trapezium be parallel; the triangle contained

by either of the other sides, and the two straight lines drawn from its

extremities to the bisection of the opposite side, is half the trapezium.

22. To divide a given trapezium, whose opposite sides are pa-

rallel, in a given ratio, by a line drawn through a given point, and

terminated by the two parallel sides.

23. If a trapezium, which has two of its adjacent angles right

angles, be bisected by a line drawn from the middle of one of those

sides which are not parallel; the sum of the parallel sides will have

to one of them the same ratio, that the side which is not bisected has

to that segment of it which is adjacent to the other.

24. If the sides of an equilateral and equiangular pentagon be

produced to meet; the angles formed by these lines are together

equal to two right angles.

25. If the sides of an equilateral and equiangular hexagon be pro-

duced to meet; the angles formed by these lines are together equal

to four right angles.

26. The area of any two parallelograms described on the two

sides of a triangle is equal to that of a parallelogram on the base,

whose side is equal and parallel to the line drawn from the vertex of

the triangle to the intersection of the two sides of the former parallel-

ograms produced to meet.

27. The perimeter of an isosceles triangle is greater than the

perimeter of a rectangular parallelogram, which is of the same alti-

tude with, and equal to the given triangle.

28. If from one of the acute angles of a right-angled triangle, a

line be drawn to the opposite side; the squares of that side and the

line so drawn are together equal to the squares of the segment ad-

jacent to the right angle and of the hypothenuse.

29. In any triangle, if a line be drawn from the vertex at right

angles to the base; the difference of the squares of the sides is equal

to the difference of the squares of the segments of the base.

30. In any triangle, if a line be drawn from the vertex bisecting

the base; the sum of the squares of the two sides of the triangle is

double the sum of the squares of the bisecting line and of half the

base.

31. If from the three angles of a triangle lines be drawn to the

points of bisection of the opposite sides; the squares of the distances

between the angles and the common intersection are together one
third of the squares of the sides of the triangle.

32. If from any point within or without any rectilineal figure,

perpendiculars be let fall on every side; the sum of the squares of

the alternate segments made by them will be equal.

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.

squares

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

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

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.