29. If from the extremities of the side of a square circles be

described with radii equal, the former to the side and the latter to

the diagonal of the square; the area of the lune so formed will be

equal to the area of the square.

30. If on the sides of a triangle inscribed in a circle, semicircles

be described; the two lunes formed thereby will together be equal

to the area of the triangle.

31. If on the two longer sides of a rectangular parallelogram as

diameters, two semicircles be described towards the same parts; the

figure contained by the two remaining sides of the parallelogram and

the two circumferences shall be equal to the parallelogram.

32. If two points be taken at equal distances from the extremities

of a quadrant, and perpendiculars be drawn from these points to the

radius; the mixtilinear space cut off, shall be equal to the sector

which stands on the arc between them.

33. If the arc of a semicircle be trisected, and from the points of

section lines be drawn to either extremity of the diameter; the

difference of the two segments thus made will be equal to the sector

which stands on either of the arcs.

34. If a straight line be placed in a circle, and on the radius

passing through one extremity, as a diameter, another circle be de-

scribed; the segments of the two circles cut off by the above straight

line will be similar, and in the ratio of four to one.

35. If on any two segments of the diameter of a semicircle semi-

circles be described; the area included between the three circum-

ferences will be equal to the area of a circle whose diameter is a

mean proportional between the segments.

:

36. If the diameter of a semicircle be divided into any number

of parts, and on them semicircles be described; their circumferences

will together be equal to the circumference of the given semicircle.

37. If two equal circles cut each other, and from either point of

section a line be drawn meeting the two circumferences; the area

cut off by the part of this line between the two circumferences will

be equal to the area of the triangle contained by that part and lines

drawn to its extremities from the other point of section.

38. If two equal circles touch each other externally, and through

the point of contact another be described with the same radius; the

area contained by the convex circumferences cut off from the touch-

ing circles, and the part of the third without them, is equal to the

area of the quadrilateral figure formed by lines drawn from the points

of intersection to the point of contact, and to the point where the

third circle is cut by a tangent drawn to the point of contact of the

two circles.

39. If a straight line be divided into any two parts, and upon

the whole and the two parts semicircles be described; and from the

point of section a perpendicular be drawn, on each side of which

circles are described touching it and the semicircles; these circles

will be equal.

SECTION IX. Page 291.

1. GIVEN one angle, a side adjacent to it, and the difference of

the other two sides; to construct the triangle.

2. Given one angle, a side opposite to it, and the difference of

the other two sides; to construct the triangle.

3. Given the base, and one of the angles at the base; to con-

struct the triangle when the side opposite to the given angle is equal

to half the sum of the other side and a given line.

4. Given the base of a right-angled triangle, and the sum of the

hypothenuse and a straight line, to which the perpendicular has

a given ratio; to construct the triangle.

5. Given the perpendicular drawn from the vertical angle to the

base, and the difference between each side and the adjacent segment

of the base made by the perpendicular; to construct the triangle.

6. Given the vertical angle, and the base; to construct the

triangle when the line drawn from the vertex cutting the base in any

given ratio, bisects the vertical angle.

7. Given the vertical angle, and one of the sides containing it;

to construct the triangle, when the line drawn from the vertex

making a given angle with the base, bisects the triangle.

8. Given one angle, a side opposite to it, and the sum of the

other two sides; to construct the triangle.

9. Given the vertical angle, the line bisecting the base, and the

angle which the bisecting line makes with the base; to construct the

triangle.

10. Given the vertical angle, the perpendicular drawn from it to

the base, and the ratio of the segments of the base made by it; to

construct the triangle.

11. Given the vertical angle, the base, and a line drawn from

either of the angles at the base to cut the opposite side in a given

ratio; to construct the triangle.

12. Given the perpendicular, the line bisecting the vertical angle,

and the line bisecting the base; to construct the triangle.

13. Given the line bisecting the vertical angle, the line bisecting

the base, and the difference of the angles at the base; to construct

the triangle.

14. Given the vertical angle, and the line drawn to the base

bisecting the angle, and the difference between the base and the sum

of the sides; to construct the triangle.

15. Given the line bisecting the vertical angle, the perpendicular

drawn to it from one of the angles at the base, and the other angle at

the base; to construct the triangle.

16. Given the line bisecting the vertical angle, and the perpen-

diculars drawn to that line from the extremities of the base; to

construct the triangle.

17. Given the vertical angle, the difference of the two sides con-

taining it, and the difference of the segments of the base made by

a perpendicular from the vertex; to construct the triangle.

18. Given the base, and vertical angle; to construct the triangle,

when the square of one side is equal to the square of the base, and

three times the square of the other side.

19. Given the base and perpendicular; to construct the triangle,

when the rectangle contained by the sides is equal to twice the

rectangle contained by the segments of the base made by the line

bisecting the vertical angle.

g

20. In a right-angled triangle, having given the sum of the base and hypothenuse, and the sum of the base and perpendicular; to construct the triangle.

21. Given the perimeter of a right-angled triangle whose sides are in geometrical progression; to construct the triangle.

22. Given the difference of the angles at the base, the ratio of the segments of the base made by the perpendicular, and the sum of the sides; to construct the triangle.

23. Given the difference of the angles at the base, the ratio of the sides, and the length of a third proportional to the difference of the segments of the base made by a perpendicular from the vertex and the shorter side; to construct the triangle.

24. Given the base of a right-angled triangle; to construct it, when parts, equal to given lines, being cut off from the hypothenuse and perpendicular, the remainders have a given ratio.

25. Given one angle of a triangle, and the sums of each of the sides containing it and the third side; to construct the triangle.

26. Given the vertical angle, and the ratio of the sides containing it, as also the diameter of the circumscribing circle; to construct the triangle.

27. Given the vertical angle, and the radii of the inscribed and circumscribing circles; to construct the triangle.

28. Given the vertical angle, the radius of the inscribed circle, and the rectangle contained by the straight lines drawn from the centre of that circle to the angles at the base; to construct the triangle.

29. Given the base, one of the angles at the base, and the point in which the diameter of the circumscribing circle drawn from the vertex meets the base; to construct the triangle.

30. Given the vertical angle, the base, and the difference between two lines drawn from the centre of the inscribed circle to the angles at the base; to construct the triangle.

31. Given that segment of the line bisecting the vertical angle which is intercepted by perpendiculars let fall upon it from the angles at the base; the ratio of the sides; and the ratio of the radius of the

inscribed circle to the segment of the base which is intercepted between the line bisecting the vertical angle and the point of contact of the inscribed circle; to construct the triangle.

32. Given the line bisecting the vertical angle, and the differences between each side and the adjacent segment of the base made by the bisecting line; to construct the triangle.

33. Given one of the angles at the base, the side opposite to it, and the rectangle contained by the base and that segment of it made by the perpendicular which is adjacent to the given angle; to construct the triangle.

34. Given the vertical angle, and the lengths of two lines drawn from the extremities of the base to the points of bisection of the sides; to construct the triangle.

35. Given the lengths of three lines drawn from the angles to the points of bisection of the opposite sides; to construct the triangle.

36. Given the segments of the base made by the perpendicular, and one of the angles at the base triple the other; to construct the triangle.

37. The area and hypothenuse of a right-angled triangle being given; to construct the triangle.

38. Given one angle, and a line drawn from one of the others bisecting the side opposite to it; to construct the triangle, when the area is also given.

39. In two similar right-angled triangles, the sum of the base of one and perpendicular of the other is given; to determine the triangles such that their hypothenuses may contain the right angle of another triangle similar to them, and the sum of the three areas may be equal to a given area.

40. Given the vertical angle, the area, and the distance between the centres of the inscribed circle and the circle which touches the base and the two sides produced; to construct the triangle.

41. Given the area, the line from the vertex dividing the base into segments which have a given ratio, and either of the angles at the base; to construct the triangle.