can be drawn to the circumference, one on each side of AO. For corresponding to D, there will be another point Hon the other side of AO, such that AHAD, and every other line would be unequal to AH, and therefore to AD. THEOREM IV. Of two straight lines AD, AE, drawn from an internal point A to the circumference, AD, which is nearer to AO, is greater than AE, which is more remote. Join OE meeting AD in F. Then FE is the shortest line that can be drawn from F to the circumference (II. 2). Therefore FD> FE: add to each AF. but Then AD> AF+FE, AF+FE> AE. Therefore AD>AE. COR. 1. From the same point A, two and only two equal straight lines can be drawn to the circumference, one on each side of AO. COR. 2. The same proof will hold good if A be taken on the circumference. E D Intersection and Contact. THEOREM V. A straight line xy cannot cut the circumference in more than two points. (Fig. on next page.) For only two equal straight lines can be drawn from the centre O to xy. [I. 6. A straight line which cuts a circle is called a secant; a straight line which meets the circumference but does not cut it is called a tangent: as DBE. A straight line drawn through the point of contact perpendicular to the tangent is called a normal. We shall find that the normal always passes through the centre. If one circumference meets another but does not cut it, it is said to touch it. THEOREM VI. The straight line AB drawn at right angles to the radius 04 from its extremity A is a tangent to the circle at A. For the perpendicular OA is shorter than any other line OC that can be drawn from 0 to A. Hence every other point in AB except A lies without the circumference. B A Conversely. Every tangent is perpendicular to the radius drawn to the point of contact. For the radius OA is the shortest line that can be drawn to AB; it is therefore perpendicular to it. COR. 1. There is but one tangent at any point. COR. 2. The normal at every point passes through the centre. THEOREM VII. The circumferences of two circles O and I cannot cut one another in more than two points; nor touch in more than one. I 0 0 B For every point on the circumference of I is equally distant from the centre I. But only two equal straight lines can be drawn from I to the circumference of 0. [II. 3. Cor. Therefore the circumference of O cannot meet that of I in more than two points. Let A and B be points where the circumference of meets that of I. Join IA, IB. Then since IA IB these lines lie on different sides of the shortest line that can be drawn to the circumference of 0, and either is greater than that line. [II. 3. Cor. II. 2. |