THEOREM 5. The angle subtended at any point in the circum- ference by any arc of a circle is half of the angle subtended COR. I. Angles in the same segment of a circle are equal to one COR. 2. If a circle is divided into any two segments by a chord, The locus of a point at which a given straight line sub- tends a constant angle is an arc of a circle. THEOREM 6. A tangent meets the circle in one point only, viz. THEOREM 7. The radius to the point of contact is at right angles THEOREM 8. If from the point of contact of a straight line and a circle a chord of the circle be drawn, the angles made by THEOREM 9. From any point within or without a circle except the centre, two and only two normals can be drawn, one of which is the shortest, and the other the longest line that can be drawn from that point to the circumference: and as a point moves along the circumference from the extremity of the shortest to the extremity of the longest normal, its distance from the fixed point continually increases THEOREM IO. . Intersection of Circles. The line that joins the . 104 104 PROBLEM 3. To cut from any circle a segment which shall be PROBLEM 4. On a given straight line to describe a segment of a PROBLEM 6. To find the locus of the centres of circles which touch two given straight lines PROBLEM 7. To describe a circle to touch three given straight THEOREM 12. If from the centre of a circle radii are drawn to make equal angles with one another consecutively all round, then if their extremities are joined consecutively, a regular polygon will be inscribed in the circle, and if at their extre- mities, tangents are drawn, a regular polygon will be cir- THEOREM 13. In a regular polygon the bisectors of the angles PROBLEM S. To construct a regular polygon of four, eight, six- To construct regular polygons of three, six, THEOREM 14. The area of a circle is equal to half the rectangle BOOK III. PROPORTION. INTRODUCTION. Measures. PROBLEM 1. To find the greatest common measure of two magni tudes, if they have a common measure THEOREM I. To prove that the side and diagonal of a square are PAGE . 125 . 126 THEOREM 2. If A and B be two fixed points in a straight line of indefinite length, and P a moveable point in that line, then 128 129 133 134 THEOREM 3. If A, B, C, D be four magnitudes such that B and D always contain the same aliquot part of A and C respectively the same number of times, however great the number of parts into which A and C are divided, then A : B :: C: D. 135 FIVE COROLLARIES 136 SECTION I. APPLICATION OF PROPORTION TO LINES. THEOREM 4. If two straight lines are cut by three parallel straight lines, the segments made on the one are in the same ratio as the segments made on the other THEOREM 5. If a line bisect the vertical angle of a triangle and meet the base, it will divide the base into two segments FIVE COROLLARIES THEOREM 6. If two triangles have two angles of the one equal respectively to two angles of the other, the triangles shall be THEOREM 7. If two triangles have one angle of the one equal to THEOREM 8. If the sides about each of the angles of two triangles are proportionals, the triangles will be similar THEOREM 9. If two triangles have the sides about an angle of the one triangle proportional to the sides about an angle of the other, and have also the angle opposite that which is not the less of the two sides of the one equal to the corresponding angle of the other, these triangles will be similar PAGE 138 139 140 141 142 144 145 146 |