BOOK II. THE CIRCLE. DEFINITIONS. 210. A circle is a portion of a plane bounded by a curved line called a circumference, all points of which are equally distant from a point within called the centre. 211. A radius is a straight line drawn from the centre to the circumference; and a diameter is a straight line drawn through the centre, having its extremities in the circumference. By the definition of a circle, all its radii are equal. All its diameters are equal, since the diameter is equal to two radii. 212. A secant is a straight line which intersects the circumference in two points; as, AD, Fig. 1. 213. A tangent is a straight line which touches the circumference but does not intersect it; as, BC, Fig. 1. The point in which the tangent touches the circumference is called the point of contact, or point of tangency. B FIG. 1. 214. Two circumferences are tangent to each other when they are both tangent to a straight line at the same point; and are tangent internally or externally, according as one circumference lies wholly within or without the other. 68. ABCD is a parallelogram, E and F the middle points of AD and BC respectively; show that BE and DF will trisect the diagonal AC. 69. If from the diagonal BD of a square ABCD, BE is cut off equal to BC, and EF is drawn perpendicular to BD, show that DE is equal to EF, and also to FC. 70. The bisector of the vertical angle A of a triangle ABC, and the bisectors of the exterior angles at the base formed by producing the sides AB and AC, meet in a point which is equidistant from the base and the sides produced. 71. If the two angles at the base of a triangle are bisected, and through the point of meeting of the bisectors a line is drawn parallel to the base, the length of this parallel between the sides is equal to the sum of the segments of the sides between the parallel and the base. 72. If one of the acute angles of a right triangle is double the other, the hypotenuse is double the shortest side. 73. The sum of the perpendiculars dropped from any point in the base of an isosceles triangle to the legs is constant, and equal to the altitude upon one of the legs. HINT. Let PD and PE be the two Is, BF the altitude upon AC. Draw PG 1 to BF, and prove the PBG and PBD equal. A E F G P 74. The sum of the perpendiculars dropped from any point within an equilateral triangle to the three sides is constant, and equal to the altitude. HINT. Draw through the point a line Il to the base, and apply Ex. 73. 75. What is the locus of all points equidistant from a pair of intersecting lines? 76. In the triangle CAB the bisector of the angle С makes with the perpendicular from C to AB an angle equal to half the difference of the angles A and B. 77. If one angle of an isosceles triangle is equal to 60°, the triangle is equilateral. BOOK II. THE CIRCLE. DEFINITIONS. 210. A circle is a portion of a plane bounded by a curved line called a circumference, all points of which are equally distant from a point within called the centre. 211. A radius is a straight line drawn from the centre to the circumference; and a diameter is a straight line drawn through the centre, having its extremities in the circumference. By the definition of a circle, all its radii are equal. All its diameters are equal, since the diameter is equal to two radii. 212. A secant is a straight line which intersects the circumference in two points; as, AD, Fig. 1. 213. A tangent is a straight line which touches the circumference but does not intersect it; as, BC, Fig. 1. The point in which the tangent touches the circumference is called the point of contact, or point of tangency. B Ө FIG. 1. 214. Two circumferences are tangent to each other when they are both tangent to a straight line at the same point; and are tangent internally or externally, according as one circumference lies wholly within or without the other. 215. An arc of a circle is any portion of the circumference. An arc equal to one-half the circumference is called a semicircumference. 216. A chord is a straight line having its extremities in the circumference. Every chord subtends two arcs whose sum is the circumference; thus, the chord AB (Fig. 3) subtends the smaller arc AB and the larger arc BCDEA. If a chord and its arc are spoken of, the less arc is meant unless it is otherwise stated. 217. A segment of a circle is a portion of a circle bounded by an arc and its chord. A segment equal to one-half the circle is called a semicircle. 218. A sector of a circle is a portion of the circle bounded by two radii and the arc which they intercept. A sector equal to one-f e-fourth of the circle is called a quadrant. 219. A straight line is inscribed in a circle if it is a chord. 220. An angle is inscribed in a circle if its vertex is in the circumference and its sides are chords. 221. An angle is inscribed in a segment if its vertex is on the arc of the segment and its sides pass through the extremities of the arc. 222. A polygon is inscribed in a circle if its sides are chords of the circle. 223. A circle is inscribed in a polygon if the circumference touches the sides of the polygon but does not intersect them, 224. A polygon is circumscribed about a circle if all the sides of the polygon are tangents to the circle. 225. A circle is circumscribed about a polygon if the circumference passes through all the vertices of the polygon. 226. Two circles are equal if they have equal radii; for they will coincide if one is applied to the other; conversely, two equal circles have equal radii. Two circles are concentric if they have the same centre. PROPOSITION I. THEOREM. 227. The diameter of a circle is greater than any other chord; and bisects the circle and the circumference. M E B P Let AB be the diameter of the circle AMBP, and AE any other chord. To prove AB> AE, and that AB bisects the circle and the circumference. Proof. I. From C, the centre of the O, draw CE. But CE=CB, (being radii of the same circle). AC+ CE > AE, (the sum of two sides of a ▲ is the third side). AC+ CB > AE, or AB > AE. § 137 Ax. 9 Then II. Fold over the segment AMB on AB as an axis until it falls upon APB, § 59. The points A and B will remain fixed; therefore the arc AMB will coincide with the arc APB; because all points in each are equally distant from the centre C. § 210 Hence the two figures coincide throughout and are equal. § 59 Q. E. D |