Page images
PDF
EPUB
[blocks in formation]

Note. Care should be taken not to confound the circle with its circumference. The pupil should bear in mind that the circle is a surface which has length and breadth, while the circumference is simply a line.

2. Every straight line, CA, CE, CD, drawn from the centre to the circumference, is called a radius or semidiameter; every line which, like AB, passes through the centre, and is terminated on both sides by the circumference, is called a diameter.

From the definition of a circle, it follows that all the radii are equal; also, that all the diameters are equal, and each is double of the radius.

3. A portion of the circumference, such as FHG, is called

an arc.

The chord or subtense of an arc is the straight line FG, which joins its two extremities.

4. A segment is the surface, or portion of a circle, included between an arc and its chord.

Note. In all cases, the same chord FG belongs to two arcs, FHG, FEG, and consequently to two segments: but the smaller one is always meant, unless the contrary is expressed.

5. A sector is the part of the circle included between an arc DE, and two radii, CD, CE, drawn to the extremities of the arc.

6. A line is said to be inscribed in a circle, when its extremities are in the circumference, as AB.

7. An inscribed angle is one which, like BAC, has its vertex in the circumference, and is formed by two chords.

C

8. An inscribed triangle is one which, like BAC, has its three angular points in the circumference.

9. And, generally, an inscribed figure is one, of which all the angles have their vertices in the circumference. The circle is said to circumscribe such a figure.

10. A secant is a straight line which cuts the circumference in two points. Thus AB is a secant.

11. A tangent is a straight line which touches the circumference, but which, when produced, does not

cut it. Thus CD is a tangent.

B

C

-D

M

The point M, is called the point of contact.

12. In like manner, two circumferences touch each other when they have but one point in common.

13. A polygon is circumscribed about a circle, when all its sides are tangents to the circumference. (See the diagram of Prop. 7. 5.) In the same case, the circle is said to be inscribed in the polygon.

5

PROPOSITION I. THEOREM.

The diameter AB, divides the circle AFB and its circumference into two equal parts.

For, if the figure AEB be applied to AFB, their common base AB retaining its position, the curve line AEB must fall exactly on the curve line AFB, A otherwise there would be points in the one or the other, unequally distant from the centre, which is contrary to the definition of a circle. Hence,

F

E

Every diameter bisects the circle and its circumference.

B

[blocks in formation]

The chord AD, is less than the diameter AB.

For, if the radii CA, CD, (see the last figure,) be drawn to the extremities of the chord AD, we shall have the straight line AD<AC+CD; (Prop. 7. 1;) consequently AD AB. Therefore,

Every chord is less than the diameter.

Cor. The greatest line which can be inscribed in a circle is equal to its diameter.

[blocks in formation]

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

For, if it could meet it in three, those three points would be equally distant from the centre; (Def. 2. 2;) hence, there would be three equal straight lines drawn from the same point

to the same straight line, which is impossible. (Prop. 15. 1. Cor. 2.) Hence, A straight line, &c.

PROPOSITION IV. THEOREM.

In the same circle, or in equal circles, equal arcs are subtended by equal chords; and, conversely, equal chords subtend equal arcs.

[blocks in formation]

For, since the diameters AB, EF are equal, the semicircle AMDB may be applied exactly to the semicircle ENGF, and the curve line AMDB will coincide entirely with the curve line ENGF. But the part AMD is equal to the part ENG by hypothesis, hence the point D will fall on G; therefore the chord AD is equal to the chord EG.

Conversely, suppose again the radii AC, EO to be equal, if the chord AD is equal to the chord EG, the arcs AMD, ENG will also be equal.

For, if the radii CD, OG be drawn, the triangles ACD, EOG, having all their sides respectively equal, namely, AC=EO, CD=OG, and AD=EG, are themselves equal; and, consequently, the angle ACD is equal EOG. (10. 1.) Now, placing the semicircle ADB on its equal EGF, since the angles ACD, EOG are equal, it is plain that the radius CD will fall on the radius OG, and the point D on the point G; therefore the arc AMD is equal to the arc ENG. Hence, In the same circle, &c.

PROPOSITION V. THEOREM.

In the same circle, or in equal circles, the greater arc is subtended by the greater chord; and, conversely, the greater chord subtends the greater arc.

Let the arc AH be greater than AD; and draw the chords AD, AH, and the radii CD, CH. The chord AH is greater than AD. For the two sides A AC, CH of the triangle ACH are equal

to the two AC, CD of the triangle

D

H

B

ACD, and the angle ACH is greater

K

than ACD; hence the side AH is greater than AD; (9. 1;) consequently the chord, which subtends the greater arc, is the greater.

Conversely, if the chord AH is greater than AD, it follows, that the arc AH is greater than AD. (Ax. 9.) Hence, In the same circle, &c.

Scholium. The arcs here treated of are each less than the semicircumference. If they were greater, the reverse property would be true; as the arcs increased, the chords would diminish, and conversely. Thus, the arc AKBD being greater than AKBH, the chord AD, of the first, is less than the chord AH of the second.

PROPOSITION VI. THEOREM.

If the radius CG, is at right-angles to a chord AB, it bisects the chord, and also the subtended arc AGB.

Draw the radii CA, CB. Then, since by hypothesis the triangles ACD, and BCD are right-angled triangles; and

« PreviousContinue »