Proof. Take O the centre; join OA, bisect it in E; and with centre E and radius EO or EA describe a circle.

B

O

Then OA will be a diameter of that circle, and each of the portions into which it divides the circumference will cut the circle BCD, because each is a continuous line with one extremity within and one extremity without the circle.

Let them meet it in T and T" respectively. Join OT and AT. Then because OTA is an angle in a semicircle it is a right angle, (III. 15.) therefore TA touches the circle BCD at the point T.

Similarly AT touches the same circle at T'.

(III. 18.)

Therefore two straight lines can be drawn from A to touch the circle.

Again, there cannot be more than two straight lines drawn from A to touch the circle.

For because the angle between the radius and the tangent is a right angle, (III. 18.) therefore the point of contact lies on the circle described on AO as diameter. (III. 16 and III. 13 Obs.)

But this circle cannot intersect the given circle in more (III. 10. Cor. 2.)

than two points.

Therefore there cannot be drawn more than two tangents from A to the circle.

COR. The two tangents drawn to a circle from an external point are equal and make equal angles with the straight line joining that point with the centre.

For let AT, AT" be the two lines touching the circle in Tand T. Then because OT is equal to OT',

and OA is common to the two triangles OẠT and OAT", and the angles at Tand 7' are right angles;

therefore the triangles are equal and the angle OAT is equal to the angle OAT';

(I. 20.)

and therefore the tangents from A are equal and make equal angles with OA.

SECTION IV. B.

TANGENTS (treated by the method of limits).

This may be omitted the first time of reading.

There is another light in which we may regard the lines of which we have been speaking in Section IV. (A), which is extremely valuable when we come to consider curves other than circles. We shall proceed to give an account of it.

Let MAN be a curve, not necessarily a circle, but one which curves in the same direction throughout as you proceed from M towards N. Take a line through A meeting the curve at some point P between A and N. Then the

nearer P is taken to A the nearer does the line AP approach to a position represented in the figure by the

line T'AT. So long as P is between A and N it can never quite coincide with the said line TAT, but it can be made to approach as near to it as we please by taking P close enough to A.

Similarly if we take a line through A meeting the curve in some point R between A and M, then the nearer that point lies to the nearer will the line AR approach the position TAT'. It can never quite coincide with the said line, as long as R is between A and M, but it may be made to approach as near to it as we please by taking R near enough to A.

It may not be easy to see how the line T'AT is to be accurately obtained, but it will easily be seen that there is in general such a line at each point of a curve, and it will be distinguished from other lines drawn through the point by the peculiarity that it does not cross the curve at that point. Such a line is said to touch the curve at that point, or, more formally, if a secant of a curve alters its position in such a manner that the two points of intersection continually approach, and ultimately coincide with one another, the secant in its limiting position is said to touch, or to he a tangent to, the curve, and the point at which the tangent meets the curve is called the point of contact.

We shall now investigate the position of the tangent to a circle at any specified point, using our newly obtained definition of a tangent.

Now let C move up towards A, then the chord AC will become shorter, and the perpendicular OE, which bisects the chord AC, will approach nearer to coincidence with OA. Hence the line AC will approach nearer to the position of being perpendicular to OA. And inasmuch as the chord AC can be made as short as we please, and thus the line OE can be made to approach as near as we please to OA, the line AK can be made to approach as near as we please to the position of AB. Hence AB is the tangent at A.

And inasmuch as no straight line can meet the circle in more than two points, and the line AB is the limiting position of a secant through A when the other point of intersection has moved up to coincidence with A, it follows that the line AB cannot meet the circle again. Hence every straight line through a point on the circumference meets it in one other point, except the straight line perpendicular to the

radius at the point, and this is the tangent at the point; which is Theorem 18.

It is evident that we shall arrive at exactly the same result by supposing that the point C is on the other side of OA, and moves up to coincidence with A in the other direction.

Def. II. If a secant of a circle alters its position in such a manner that the two points of intersection continually approach, and ultimately coincide with one another, the secant in its limiting position is said to touch or to be a tangent to, the circle.

Def. 12. The point in which two points of intersection ultimately coincide is called the point of contact and the tangent is said to touch the circle at that point.

Taking this definition of a tangent, Theorem 6 gives us

The straight line drawn from the centre to the point of contact of a tangent is perpendicular to the tangent. This is (6) in the last section:

Theorem 7 gives us

The straight line drawn from the centre perpendicular to a tangent passes through the point of contact. This is (d) in the last section.

Theorem 8 gives us

The straight line drawn perpendicular to a tangent through its point of contact passes through the centre. This is (c) in the last section.

Theorem 17, Cor. 1, gives us Theorem 19.

For if ABDC be a quadrilateral inscribed in a circle and AC be produced to K, then the angle KCD is equal