therefore L is on the circumference ;

III. 1, Cor. that is, the straight line GH meets the circumference in a second point L.

Next, let the point L move along the circumference of the circle so as to continually approach to, and ultimately coincide with, the point A :

then shall GH continually approach to, and ultimately coincide with, the perpendicular through A to OA.

As the point L moves along the circumference of the circle and continually approaches to A, the minor arc AL continually diminishes,

and therefore the angle AOL continually diminishes,

III. 5.

and when L ultimately coincides with A the angle AOL vanishes ; therefore the angle AOK, which is half the angle AOL, continually diminishes, and ultimately vanishes.

But the angle OAK is the complement of the angle AOK;

I. 25, Cor. therefore the angle OAK continually increases, and is ultimately a right angle;

that is, GH continually approaches to, and ultimately coincides with, the perpendicular through A to OA.

Q.E.D.

DEF. 13. Aliter. 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. The point in which the two points of intersection ultimately coincide is called the point of contact.

Theor. 20 may be restated as in the first two of the following Corollaries :

COR. I.

One and only one tangent can be drawn to a circle at a given point on the circumference.

COR. 2. Any tangent to a circle is perpendicular to the radius drawn to the point of contact.

COR. 3. The centre of a circle lies in the perpendicular to any tangent at the point of contact.

For only one perpendicular can be drawn to a tangent from the point of contact.

COR. 4. The straight line drawn from the centre perpendicular to the tangent meets it in the point of contact.

For only one perpendicular can be drawn from the centre to the tangent.

COR. 5. PROB. To draw a tangent to a given circle at a given point in the circumference.

Ex. 76. To draw a tangent to a given circle making a given angle with a given straight line.

Ex. 77. All equal chords in a circle may be touched by another circle.

THEOR. 21. A straight line cuts a circle, touches it, or does not meet it at all, according as its distance from the centre is less than, equal to, or greater than, the radius.

Let the distance from the centre be less than the radius,

then the foot of the perpendicular from the centre on the straight line is within the circumference ;

III. 1, Cor. also, a point may be taken on the straight line as far from the centre as we please,

and therefore without the circumference;

III. 1, Cor. that is, there are points on the straight line both within and without the circumference;

therefore the straight line cuts the circumference.

Next, let the distance from the centre be equal to the radius, then the straight line is the perpendicular to a radius at its extremity,

and therefore touches the circle.

III. 20.

Again, let the distance from the centre be greater than the radius,

then every point on the straight line is at a distance from the centre greater than the radius,

I. 15.

III. 1, Cor.

and therefore without the circumference;
therefore the straight line does not meet the circle at all.

Q.E.D.

COR. The distance of a straight line from the centre of a circle is less than, equal to, or greater than, the radius, according as the straight line cuts touches, or does not meet, the circle.

THEOR. 22. Two tangents and two only can be drawn to e circle from an external point,

Let ABC be the given circle, D the external point:

B

two tangents and two only can be drawn from D to the circle ABC.

Take O the centre of the circle ABC;

join OD, and on OD as diameter describe a circle ;

this circle will cut the circle ABC in two points, since O is

within and D is without the circle ABC,

and in two points only.

Let A and B be these points;

join DA, DB;

III. 12, Cor. 2.

DA and DB shall be the tangents drawn from D to the circle ABC.

Join OA, OB.

The angle DAO in the semicircle DAO is a right angle;

III. 17.

therefore DA touches the circle ABC.

III. 20

In like manner BD touches the circle.

Again, any other straight line drawn from D to meet the

circle ABC, as DE, is not a tangent to ABC, since the angle DEO is greater than, or less than, a right angle, according as E is within, or without the circle ABD. III. 16, Cor. 1.

Q.E.D.

COR. 1. The two tangents drawn to a circle from an external point are equal, and make equal angles with the straight line joining that point and the centre.

I. 20, Cor. I.

COR. 2. PROB. To draw a tangent to a given circle from a given point outside the circumference.

The construction and demonstration are contained in the Theorem.

*Ex. 78. If a circle touch each of two intersecting straight lines, its centre lies on one or other of the bisectors of the angles contained by these lines.

* Ex. 79. If a circle touch each of two parallel straight lines, its centre lies on the line parallel to both and midway between them.

Ex. 80. A circle is inscribed in a right-angled triangle. Shew that the hypotenuse together with the diameter is equal to the sum of the other two sides.

*Ex. 81. If a quadrilateral be circumscribed about a circle the sum of two opposite sides is equal to that of the other two opposite sides.

* Ex. 82. If a convex quadrilateral be such that the sum of