Of the Circle.

20. If from the centre of a circle a line be drawn perpendicular to a chord, it will bisect the

chord, and also the arc of the chord. Thus, CFE drawn

from the centre C, perpendic- A

ular to AB, bisects AB at F,

and also makes AE-EB.

E

B

21. The distance from the centre of a circle to a chord,

is measured on a perpendicular to the chord.

22. In the same, or in equal circles, chords which are equally distant from the centre, are equal. Thus, if CA=CB, then will the chord FG chord DE.

=

F

B

E

23. If the chord DG is equal to the chord FH, they will be equally distant from the centre: that is, CB will be equal to CA.

H

G

F D

B

QUEST.-20. If from the centre of a circle a line be drawn perpendicular to a chord, how will it divide the chord? How will it divide the arc of the chord? 21. How is the distance from the centre of a circle, to a chord measured? 22. If chords are equally distant from the centre of a circle, will they be equal or unequal? 23. If two chords are equal to each other, will they be equally or unequally distant from the centre ?

Of the Circle.

24. If several lines be drawn within a circle, the greatest is the diameter, and those nearest the centre are greater than those more re

mote.

25. There is no point except the centre, from which three equal lines can be drawn to the circumference.

26. A straight line cannot cut the circumference of a circle in more than two points.

27. Two circumferences cannot cut each other in more than two points.

QUEST.-24. If several lines are drawn within a circle, which is the greatest? Which is the least? 25. What point is that within a circle from which three or more equal lines may be drawn to the circumference? Is there any other such point? 26. In how many points can a straight line cut the circumference of a circle? 27. In how many points can the circumferences of two circles intersect each other?

Of the Circle.

28. If two circles touch each other externally, their centres and the point of contact are in the same straight line. Thus, the centres C and F, and the point of contact D are in the same straight line CDF.

29. If two circles touch each other internally, their centres and point of contact are in the same straight line. Thus, the centres F and C, and the point of contact D, are in the same straight line.

30. If two chords intersect each other the product, or rectangle of the parts of the one, is equal to the rectangle of the parts of the other. Thus, the two chords AB and CD, which intersect each other at E, give

AEX EB CEXED.

F

C

E

QUEST.-28. If two circles touch each other externally, and the centres be joined by a straight line, will this line also contain the point of contact? 29. If two circles touch each other internally, and the centres be joined, will the line contain the point of contact? 30. If two chords intersect each other, how will the rectangle of the parts of the one compare with the rectangle of the parts of the other?

Of the Circle.

31. If from a point without a circle, a tangent AB, be drawn and also a secant ACD, then will the square of the tangent be equal to the rectangle of the parts of the secant. That is,

32. If from any point, as A, without the circumference of a circle, two tan

B

gents AB, AF be drawn, these tangents F

B

will be equal to each other, and the

line ACD passing through the centre of the circle, will bisect the angle included between them.

33. If a quadrilateral be inscribed in a circle, the sum of either two of its opposite angles is equal to two right D

angles. Thus, in the quadrilateral ABCD, we have

A+C=180°, and B+D=180°

B

QUEST.-31. If from a point without a circle, you draw a tangent, and also a secant line, what will the square of the tangent be equal to? 32. If two tangent lines be drawn, how will they compare with each other? If from the same point, a line be drawn through the centre of the circle, how will it divide the angle included between the tangents? 33. If a quadrilateral be inscribed in a circle, what is the sum of either two of the opposite angles equal to ?

Of the Circle.

34. If the radius of a circle be applied six times as a chord, it will reach exactly round the circumference. Thus, if you take the radius CA and lay it off from A to B, then from B to D, &c., you will find that after laying it off six

B

times, you will exactly reach to the point A. Hence, the side of a regular hexagon (; IV. Art. 5), is equal to the radius of the circumscribing circle.

AB by D, and the circumference of the circle by C, and the diameter CD by d, and the circumference by c; we shall have

D: d :: C: C.

36. The circumference of a circle is a little more than three times as great as the diameter. If the diameter is 1, the circumference will be 3,1416.

QUEST.-34. If the radius of a circle be applied as a chord, how many radii will go round the circumference? To what then is the side of a regular hexagon equal? 35. How are the circumferences of circles to each other? 36. How many times is the circumference of a circle greater than its diameter? If the diameter is 1 what is the circumference?