THEOR. 8. The straight line drawn perpendicular to a chord through its middle point passes through the centre. the other two follow by the Rule of Identity.] two given points is the straight line that bisects at right angles the line joining those points. THEOR. 9. A straight line cannot meet the circumference of a circle in more than two points. COR. A chord of a circle lies wholly within the circle. OBS. Hence the circumference of a circle is everywhere concave towards the centre. THEOR. 10. There is one circle and only one whose circumference passes through three given points not in the same straight line. [By Loci § 3, i.] COR. I. Two circles whose circumferences have three points in common coincide wholly. COR. 2. The circumferences of two different circles cannot meet one another in more than two points. COR. 3. If from any point within a circle more than two straight lines drawn to the circumference are equal, that point is the centre, THEOR. II. In the same circle, or in equal circles, equal chords are equally distant from the centre; and of two, unequal chords the greater is nearer to the centre than the less. [First part by Theor, 7 and Book I, Theor. 20; second part by placing the chords so as to have a common extremity and using Theor. 5 and Book I, Theor. 19.] THEOR. 12. In the same circle, or in equal circles, chords that are equally distant from the centre are equal; and of two 30 A SYLLABUS OF chords unequally distant, the one nearer to the centre is the greater. [By Rule of Conversion. ] COR. The diameter is the greatest chord in a circle. SECTION 3. ANGLES IN SEGMENTS. DEF. 8. The angle formed by any two chords drawn from a point on the circumference of a circle is called an angle at the circumference, and is said to stand upon the arc between its arms. DEF. 9. An angle contained by two straight lines drawn from a point in the arc of a segment to the extremities of the chord is called an angle in the segment. THEOR. 13. An angle at the circumference is half the angle at the centre standing on the same arc. [One proof for angles of all sizes: 7 THEOR. 14. Angles in the same segment are equal to one another. [By Theor. 13.] COR. The angle subtended by the chord of a segment at a point within it is greater than, and at a point outside Book I, Theor. 9.1 line at which that line subtends a constant angle is an arc of which that line is the chord. THEOR. 15. The angle in a segment is greater than, equal to, or less than a right angle, according as the segment is less than, equal to, or greater than a semicircle. [By Theor. 13.] THEOR. 16. A segment is less than, equal to, or greater than a semicircle, according as the angle in it is greater than, equal to, or less than a right angle. [By Rule of Conversion.] THEOR. 17. The opposite angles of a quadrilateral inscribed in a circle are supplementary. [By Theor. 13.] COR. I. Each exterior angle of a quadrilateral inscribed in a circle is equal to the interior angle whose vertex is opposite to its own. COR. 2. If the opposite angles of a quadrilateral are supple mentary, the quadrilateral can be inscribed in a circle. SECTION 4. A. TANGENTS (treated directly). DEF. 10. A secant is a straight line of unlimited length which meets the circumference of a circle in two points. THEOR. 18. Every straight line through a point on the circum ference meets it in one other point, except the straight line perpendicular to the radius at the point. [By Book I, Theor. 19.] DeF. II. A straight line which, though produced indefinitely, meets the circumference of a circle in one point only is said to touch, or to be a tangent to, the circle. DEF. 12. The point at which a tangent meets the circumference is called the point of contact. The following are immediate consequences of Theorem 18. (a) One and only one tangent can be drawn to a circle at a given point on the circumference. (6) The tangent to a circle is perpendicular to the radius drawn to the point of contact. 32 A SYLLABUS OF (C) The centre of a circle lies in the perpendicular to the tangent at the point of contact. (d) The straight line drawn from the centre perpendicular to the tangent passes through the point of contact. OBs. On the relative position of a straight line and a circle. the Rule of Conversion. THEOR. 19. Each angle contained by a tangent and a chord drawn from the point of contact is equal to the angle in the alternate segment of the circle. [Euclid's proof.] THEOR. 20. Two tangents and two only can be drawn to a circle from an external point. [By Theor. 15 and Theor. 14, Cor.] 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 and the centre. [By Book I, Theor. 20, Cor.] SECTION 4. B. TANGENTS (treated by the method of Limits). [NOTE.—The Theorems of this Section have been arranged so as to correspond with those of Section 4, A. Each Section is complete in itself.] DEF. 10. As in 4. A. DEF. Il. 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 the two points of intersection ultimately coincide is called the point of contact and the tangent is said to touch the circle at that point. Consequences (a) () (c) (d) as in 4.. A. OBs. On the relative position of a straight line and a circle. As in 4. A. COR. Enunciation and Proof as in 4. A. SECTION 5. Two CIRCLES. THEOR. 21. The straight line which passes through the centres of two circles whose circumferences meet in two points bisects the straight line joining those points, and is at right angles to it. [By Book II, Theors. 15 and 5; or by Loci § 2, iii.] THEOR. 22. If the circumferences of two circles meet at a point on the straight line passing through their centres, these circumferences cannot have a second point in common. [Contrapositive of part of Theor. 21.) DEF. 13. Two circles whose circumferences meet in one point only are said to touch each other, and the point at which they meet is called their point of contact. THEOR. 23. If the circumferences of two circles have one common point not on the line through their centres, they have |