34 A SYLLABUS OF also another common point. [Observe Theor. 22. Direct geometrical proof.] THEOR. 24. If two circles touch one another, the line through their centres passes through their point of contact. [Con trapositive of Theor. 23.) COR. Two circles that touch one another have a common tangent at the point of contact. [By Theor. 18.] OBS. (1). If the distance between the centres of two circles is greater than the sum of their radii, their circumferences will not meet and each circle will be wholly outside the other. OBS. (2). If the distance between the centres of two circles is equal to the sum of their radii, their circumferences will meet in one point only, and each circle will lie outside the other. DEF. 14. In this case the circles are said to touch externally. OBS. (3). If the distance between the centres of two circles is less than the sum and greater than the difference of their radii, their circumferences will meet in two points. DEF. 15. In this case the circles are said to cut one another. OBS. (4). If the distance between the centres of two circles is equal to the difference of their radii, their circumferences will meet in one point only, and one circle will lie within the other. DEF. 16. In this case the circles are said to touch internally. OBS. (5). If the distance between the centres of the two circles is less than the difference of their radii, their circumferences will not meet and one circle will be wholly within the other. OBS. (6). The converse of each of the above five Theorems is true. [Rule of Conversion.] SECTION 6. PROBLEMS. PROB. I. To find the centre of a given circle, or of a given arc. PROB. 2. To bisect a given arc. or outside the circumference: Discussion on the number of common tangents that can be drawn to two circles according to the relative position of the circles. PROB. 5. To describe a circle passing through three given points which are not in the same straight line. PROB. 6. To describe a circle touching three given straight lines which are not all parallel and do not all pass through the same point. Def. 17. A circle that touches the three sides of a triangle is called an inscribed circle. DEF. 18. A circle that touches one side of a triangle and the other two sides produced is called an escribed circle. Discussion on the inscribed and escribed circles of a triangle. PROB. 7. In a given circle to inscribe a triangle equiangular to a given triangle. PROB. 8. About a given circle to circumscribe a triangle equi angular to a given triangle. PROB. 9. On a given straight line to describe a segment of a circle containing a given angle. PROB. 10. From a given circle to cut off a segment containing a given angle. THEOR. 25. If the whole circumference of a circle is divided into any number of equal arcs, the inscribed polygon formed by the chords of these arcs is regular; and the circumscribed polygon formed by tangents drawn at all the points of division is also regular. THEOR. 26. If straight lines are drawn bisecting two angles of a regular polygon, the point in which the bisectors intersect is equidistant from all the vertices of the polygon and from all the sides. PROB. II. To inscribe a circle in, or to circumscribe one about, a given regular figure. PROB. 12. To inscribe in, or to circumscribe about, a given circle regular figures of 4, 8, 16, 32 .... sides. PROB. 13. To inscribe in, or to circumscribe about, a given circle regular figures of 3, 6, 12, 24.... sides. SECTION 7. THE CIRCLE IN CONNECTION WITH AREAS. THEOR. 27. If a chord of a circle is divided into two segments by a point in the chord or in the chord produced, the rectangle contained by these segments is equal to the difference of the squares on the radius and on the line joining the given point with the centre of the circle. COR. I. The rectangle contained by the segments of any chord passing through a given point is the same, whatever be the direction of the chord. COR. 2. If the point is within the circle, the rectangle contained by the segments of any chord passing through it is equal to the square on half that chord which is bi sected by the given point. COR. 3. If the point is without the circle, the rectangle con tained by the segments of any chord passing through it is equal to the square on the tangent to the circle drawn from that point. COR. 4. Conversely, if the rectangle contained by the segments of a chord passing through an external point is equal to the square on a line joining that point to a point in the circumference of the circle, this line touches the circle. PROB. 14. To inscribe in a circle a regular decagon; and thence to circumscribe a regular decagon about a circle; also to inscribe in, or to circumscribe about, a given circle a regular pentagon, or regular figures of 20, 40, 80.... sides. PROB. 15. To inscribe in a circle a regular quindecagon ; and thence to circumscribe a regular quindecagon about BOOK IV. FUNDAMENTAL PROPOSITIONS OF PROPORTION. SECTION 1. OF RATIO AND PROPORTION. [Although the Association regards a complete treatment of Proportion, such as that contained in this Book, as indispensable to a sound knowledge of Geometry, Book V may be read immediately after Book III by students who are acquainted with the treatment of Ratio and Proportion given in books on Arithmetic and Algebra.] [Notation. In what follows, large Roman letters, A, B, etc., are used to denote magnitudes, and where the pairs of magnitudes compared are both of the same kind they are denoted by letters taken from the early part of the alphabet, as A, B compared with C, D; but where they are or may be of different kinds, from different parts of the alphabet, as A, B compared with P, Q or X, Y. Small Italic letters, m, n, etc., denote whole numbers. By m. A or mA is denoted the mth multiple of A and it may be read as m times A. The product of the numbers m and n is denoted by mu, and it is assumed that mn = nm. The combination m.nA denotes the mth multiple of the nth multiple of A and may be read as m times nA, and mnA or mn . A as mn times A. By (m + n) A is denoted m + n times A.] DEF. I. One magnitude is said to be a multiple of another magnitude when the former contains the latter an |