PART III.

THE CIRCLE.

DEFINITIONS AND FIRST PRINCIPLES.

1. A circle is a plane figure contained by a line traced out by a point which moves so that its distance from a certain fixed point is always the same.

The fixed point is called the centre, and the bounding line is called the circumference.

NOTE. According to this definition the term circle strictly applies to the figure contained by the circumference; it is often used however for the circumference itself when no confusion is likely to arise.

2. A radius of a circle is a straight line drawn from the centre to the circumference. It follows that all radii of a circle are equal.

3. A diameter of a circle is a straight line drawn through, the centre and terminated both ways by the circumference.

4. A semi-circle is the figure bounded by a diameter of a circle and the part of the circumference cut off by the diameter.

It will be proved on page 142 that a diameter divides a circle into two identically equal parts.

5. Circles that have the same centre are said to be concentric.

From these definitions we draw the following inferences :

(i) A circle is a closed curve; so that if the circumference is crossed by a straight line, this line if produced will cross the circumference at a second point.

(ii) The distance of a point from the centre of a circle is greater or less than the radius according as the point is without or within the circumference.

(iii) A point is outside or inside a circle according as its distance from the centre is greater or less than the radius.

(iv) Circles of equal radii are identically equal. For by superposition of one centre on the other the circumferences must coincide at every point.

(v) Concentric circles of unequal radii cannot intersect, for the distance from the centre of every point on the smaller circle is less than the radius of the larger.

(vi) If the circumferences of two circles have a common point they cannot have the same centre, unless they coincide altogether.

6. An arc of a circle is any part of the circumference.

7. A chord of a circle is a straight line joining any two points on the circumference.

NOTE. From these definitions it may be seen that a chord of a circle, which does not pass through the centre, divides the circumference into two unequal arcs; of these, the greater is called the major arc, and the less the minor arc. Thus the major arc is greater, and the minor arc less than the semicircumference.

The major and minor arcs, into which a circumference is divided by a chord, are said to be conjugate to one another.

SYMMETRY.

Some elementary properties of circles are easily proved by considerations of symmetry. For convenience the definition given on page 21 is here repeated.

DEFINITION 1. A figure is said to be symmetrical about a line when, on being folded about that line, the parts of the figure on each side of it can be brought into coincidence.

The straight line is called an axis of symmetry.

That this may be possible, it is clear that the two parts of the figure must have the same size and shape, and must be similarly placed with regard to the axis.

DEFINITION 2. Let AB be a straight line and P a point outside it.

From P draw PM perp. to AB, and produce it to Q, making MQ equal to PM.

Then if the figure is folded about AB, the point P may be made to coincide with Q, for the LAMP = the LAMQ, and MP= MQ.

The points P and Q are said to be symmetrically opposite with regard to the axis AB, and each point is said to be the image of the other in the axis.

NOTE. A point and its image are equidistant from every point on the axis. See Prob. 14, page 91.

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SOME, SYMMETRICAL PROPERTIES OF CIRCLES.

I. A circle is symmetrical about any diameter.

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Let APBQ be a circle of which O is the centre, and AB any diameter.

It is required to prove that the circle is symmetrical about AB. Proof. Let OP and OQ be two radii making any equal LAOP, AOQ on opposite sides of OA.

Then if the figure is folded about AB, OP may be made to fall along OQ, since the AOP = the AOQ.

And thus P will coincide with Q, since OP=OQ.

Thus every point in the arc APB must coincide with some point in the arc AQB; that is, the two parts of the circumference on each side of AB can be made to coincide.

.. the circle is symmetrical about the diameter AB.

COROLLARY. If PQ is drawn cutting AB at M, then on folding the figure about AB, since P falls on Q, MP will coincide with MQ,

and the OMP will coincide with the LOMQ,

.. these angles, being adjacent, are rt. 4*;

the points P and Q are symmetrically opposite with

regard to AB.

Hence, conversely, if a circle passes through a given point P, it also passes through the symmetrically opposite point with regard to any diameter.

DEFINITION. The straight line passing through the centres of two circles is called the line of centres.

II. Two circles are divided symmetrically by their line of centres.

Let O, O' be the centres of two circles, and let the st. line through O, O' cut the Oces at A, B and A', B'. Then AB and A'B' are diameters and therefore axes of symmetry of their respective circles. That is, the line of centres divides each circle symmetrically.

III. If two circles cut at one point, they must also cut at a second point; and the common chord is bisected at right angles by the line of centres.

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Let the circles whose centres are O, O' cut at the point P. Draw PR perp. to OO', and produce it to Q, so that RQ = RP.

Then P and Q are symmetrically opposite points with regard to the line of centres O0′;

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.. since P is on the Oe of both circles, it follows that Q is also on the Oce of both. [I. Cor.]

And, by construction, the common chord PQ is bisected at right angles by 00'.