two right angles. There are thus two angles AOQ, one taken in the direction of the arrow greater than two right angles, and the other taken in the opposite direction less than two right angles.

Thus, in the figure, the angle BOA standing on the arc A CB is less than two right angles; and the angle AOB on the arc ADB is Ap greater than two right angles.

Hence it is clear that the angle at the centre standing on any arc is less than, equal to, or greater than two right angles

D

according as the arc is less than, equal to, or greater than a semi-circumference.

COR. 2. If a circle is divided into any two segments by a chord, the angles in the segments will be supplementary to one another.

For of the two angles at O, the one is double of the angle in the segment ADB, and the other of the angle in the segment ACB.

But the angles at O make up four right angles, therefore the angles in the segments ACB, ADB make up two right angles.

These segments are called Supplementary Segments.

D

COR. 3. The angle in a semicircle is a right angle.

B

For let APB be a semicircle: then the angle in the segment is half the angle at the centre: that is, the angle APB is half the angle AOB, which is two right angles; therefore the angle APB is a right angle.

Def. 13. A polygon is said to

B

be inscribed in a circle, when its angular points are on the circumference of the circle.

COR. 4.

The opposite angles of every quadrilateral figure

inscribed in a circle are together equal

to two right angles.

For the segments ABC, ADC are supplementary segments, and so also are the segments BAD, BCD.

Therefore by Cor. 3 the angles ABC, ADC are together equal to two right angles; and the angles BAD, BCD are also together equal to two right angles.

B

COR. 5. The locus of a point at which a given straight line subtends a constant angle is an arc of a circle. For if AB be the given straight

line, and on it there is described a segment capable of the given angle, at every point in the arc the straight line subtends the given angle, and at every other point the angle is greater or less, according as it is within or without the segment.

B

The segment may be described on both sides of the line AB.

Cor.

It will be seen that this is the converse and opposite of

I.

COR. 6. In equal circles equal angles at the circumferences stand upon equal arcs, and conversely.

This may be proved by superposition.

EXERCISES.

I. Prove that the lines which join the extremities of equal arcs in a circle are either equal or parallel.

2. If two opposite angles of a quadrilateral figure are together equal to two right angles, prove that a circle which passes through three of its angular points will also pass through the fourth.

3. If two opposite sides of a quadrilateral inscribed in à circle are equal, prove that the other two are parallel.

4. AB, CD are chords of a circle which cut at a constant angle. Prove that the sum of the arcs AC, BD remains constant, whatever may be the position of the chords.

5. If the diameter of a circle be one of the equal sides of an isosceles triangle, prove that its circumference will bisect the base of the triangle.

6. Circles are described on two sides of a triangle as diameters. Prove that they will intersect on the third side

or third side produced.

7. Any number of chords of a circle are drawn through a point on its circumference: find the locus of their middle points.

8. If through any point, within or without a circle, lines are drawn to cut the circle, prove that the locus of the middle points of the chords so formed is a circle.

SECTION III.

THE TANGENT AND NORMAL.

Def. 14. When a straight line cuts a circle it is called a secant.

Thus ABCD is a secant of the circle ACE.

E

T

Def. 15. When one of the points in which a secant cuts a circle is made to move up to, and ultimately coincide with the other, the ultimate position of the secant is called the tangent at that point.

Thus, if ABCD be conceived to revolve round B, in the direction of the arrow, the point C will move to C' and will ultimately coincide with B, and the line TBS, which is the position the secant then attains, is said to be a tangent to the circle at the point B.

The point B is then called the point of contact.

Several important properties follow at once from this definition of a tangent.

THEOREM 6.

A tangent meets the circle in one point only, viz. the point of contact.

For since a secant can cut a circle in two points only, it follows that the parts AB, CD are wholly without the circle; and therefore when C moves up to B, and the chord BC is merged in the point B, the whole line, with exception of the point B, is outside the circle.

THEOREM 7.

The radius to the point of contact is at right angles to the tangent.

For if F be the middle point of the chord BC, OF is perpendicular to BC; and as C moves to B, F will also move up to B, and when the secant becomes a tangent, OF, which is always at right angles to the secant, coincides with the radius OB.

Therefore OB is at right angles to the tangent TBS.

COR. I. Hence there can be only one tangent to a circle at a given point.

COR. 2, The line at right angles to the tangent through the point of contact passes through the centre.

Def. 16. When a secant is drawn from the point of contact of a tangent it divides the circle into segments which are said to be alternate to the angles made by the tangent with the secant on its sides opposite to the segments.