by this line produced, and one of the others is equal to the angle

contained by the third and a perpendicular drawn from the common

point of intersection of the three lines, to the aforesaid side.

31. In a right-angled triangle, if a straight line be drawn parallel

to the hypothenuse, and cutting the perpendicular drawn from the

right angle; and through the point of intersection a line be drawn

from one of the acute angles to the opposite side, and the extremity

of this line and of the perpendicular be joined; the locus of its

intersection with the line parallel to the hypothenuse will be a straight

line.

32. If from the angles of a triangle, lines, each equal to a given

line, be drawn to the opposite sides (produced if necessary); and

from any point within, lines be drawn parallel to these, and meeting

the sides of the triangle; these lines will together be equal to the

given line.

33. If the sides of a triangle be cut proportionally, and lines be

drawn from the points of section to the opposite angles; the inter-

sections of these lines will be in the same line, viz. that drawn from

the vertex to the middle of the base.

34. If from any point in one side of a triangle, two lines be drawn,

one to the opposite angle, and the other parallel to the base, and the

former intersect a line drawn from the vertex bisecting the base;

this point of intersection, that of the line parallel to the base and the

third side, and the third angular point are in the same straight line.

35. If one side of a triangle be divided into any two parts, and

from the point of section two straight lines be drawn parallel to, and

terminating at the other sides, and the points of termination be

joined; and any other line be drawn parallel to either of the two

former lines, so as to intersect the other, and to terminate in the

sides of the triangle; then the two extreme parts of the three seg-

ments into which the line so drawn is divided, will always be in the

ratio of the segments of the first divided line.

36. If through the point of bisection of the base of a triangle,

any line be drawn intersecting one side of the triangle and the other

produced, and meeting a parallel to the base from the vertex; this

line will be cut harmonically.

37. If from either angle of a triangle, a line be drawn inter-

secting that which joins the vertex and the bisection of the base, the

opposite side, and the line from the vertex parallel to the base; it

will be cut harmonically.

38. To draw a line from one of the angles at the base of a tri-

angle, so that the part of it cut off by a line drawn from the vertex

parallel to the base, may have a given ratio to the part cut off by the

opposite side.

39. To determine that point in the base produced of a right-

angled triangle, from which the line drawn to the angle opposite to

the base shall have the same ratio to the base produced, which the

perpendicular has to the base itself.

40. If the base of any triangle be divided into two parts by a

line which is a mean proportional between them, and which being

drawn parallel to the second side is terminated in the third; any

line parallel to the base will be divided by the mean proportional

(produced if necessary) into segments which will be to each other

inversely as the whole mean proportional to that segment which is

terminated in the third side of the triangle.

41. If from the extremities of the base of any triangle, two

straight lines be drawn intersecting each other in the perpendicular,

and terminating in the opposite sides; straight lines drawn from

thence to the intersection of the perpendicular with the base, will

make equal angles with the base.

42. In any triangle, the intersection of the perpendiculars drawn

from the angles to the opposite sides, the intersection of the lines

from the angles to the middle of the opposite sides, and the inter-

section of the perpendiculars from the middle of the sides, are all in

the same straight line. And the distances of those points from one

another are in a given ratio.

43. If straight lines be drawn from the angles of a triangle

through any point, either within or without the triangle, to meet the

sides, and the lines joining these points of intersection and the sides

of the triangle be produced to meet; the three points of concourse

will be in the same straight line.

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.

31

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.