By combining these data in different ways, many dif ferent theorems may be made.

EXERCISES.

I. If a straight line cut two concentric circles, the parts of it intercepted between the two circumferences will be equal.

2.

Of two angles at the centre, the greater angle is subtended by the greater arc; and also by the greater chord, if the sum of the two angles is less than four right angles. Prove this and its converse propositions.

3. Perpendiculars are let fall from the extremities of a diameter on any chord, or any chord produced; shew that the feet of the perpendiculars are equally distant from the

centre.

4. The locus of the points of bisection of parallel chords of a circle is the diameter at right angles to those chords.

5. If a diameter of a circle bisects a chord which does not pass through the centre, it will bisect all chords which are parallel to it.

6. AB and CD are unequal parallel chords in a circle, prove that AC and BD, and likewise AD and BC intersect on the diameter perpendicular to AB and CD, or that diameter produced, and are equally inclined to that dia

meter.

What will be the case if AB and CD are equal?

THEOREM 3.

One circle, and only one circle, can be drawn to pass through three given points which are not in the same straight line.

Let A, B, C be the three given points. Join AB, BC.

Then since AB is to be a chord, the locus of the centre is the straight line that bisects AB at right angles (II. 2, B).

Similarly, the line that bisects BC at right angles must pass through the centre. Hence the centre must be at O, the point of intersection of these perpendiculars; and the circle described with centre O and radius OA will pass through A, B and C.

And there can be only one centre, since the perpendiculars intersect in only one point.

The three points thus determine the circle.

COR. I.

cide wholly.

Circles that have three points in common, coin

Hence a circle is named by the letters which mark three points on its circumference.

COR. 2. Different circles can intersect in two points only. Def. 10. The circle is said to be circumscribed about the triangle ABC, and the triangle ABC is said to be inscribed in the circle, when the points A, B, C are on the circumference of the circle.

Def. 11. The distance of a chord from the centre is the perpendicular on the chord from the centre.

THEOREM 4.

Equal chords of a circle are equally distant from the centre, and conversely; and of two unequal chords the greater is nearer to the centre than the less, and conversely.

Let AB, CD be chords of a circle, OM, ON the perpendiculars on them from the centre, bisecting the chords in M and N respectively. Join OC, OA.

Then since M and N are right angles, therefore

and

but

therefore

D

M

N

OM2 + MC2= OC2 (1. 29),

ON2 + NA2 = OA3,

OC= OA, and OC2 = OA2;

OM2 + MC2 = ON2 + NA2.

Hence, (1) if AB = CD and AN= CM,

COR. I.

COR. 2.

AN> CM and AB> CD.

The diameter is the greatest chord of a circle.

The locus of the middle points of equal chords

in a circle is a concentric circle.

EXERCISES.

I. Given a triangle ABC to find the centre of the cir-. cumscribing circle.

2. A chord 8 inches long is drawn in a circle whose radius is 5 inches; find the distance of the chord from the

centre.

3. A chord is drawn at the distance of one foot from the centre of a circle whose diameter is 26 inches; find the length of the chord.

4. Given a circle to find its centre.

5. If two equal chords intersect one another, the segments of the one are equal to the segments of the other respectively.

6. Two chords cannot bisect one another unless both pass through the centre.

7. Given a curve, to ascertain whether it is an arc of a circle or not.

SECTION II.

ANGLES IN SEGMENTS OF THE CIRCLE.

THEOREM 5.

The angle subtended at any point in the circumference by any arc of a circle is half of the angle subtended by the same arc at the centre.

Let AB be any arc, O the centre, P any point on the

circumference. Then will the angle AOB be double of the angle APB.

Join PO, and produce it to Q.

Then because, from the definition of a circle, OPA is an isosceles triangle, the angle OAP= the angle OPA: but the exterior angle AOQ is equal to the two interior and opposite angles OAP and OPA (1. 7); and therefore the angle AOQ is double of the angle OPA. Similarly the angle QOB is double of the angle OPB.

Hence (in fig. 1) the sum, or (in fig. 2) the difference of the angles AOQ, QOB is double of the sum or dif