(b) Lay off a (or b) on c, and prove that the line representing the difference equals b or (a). Ex. 284. The sum of the perpendiculars dropped from any point in the base of an isosceles triangle to the arms, is equal to the altitude upon one of the arms. Ex. 285. The sum of the three perpendiculars dropped from any point of an equilateral triangle upon the sides is constant, and equal to the altitude of the triangle. (Ex. 284.) Ex. 286. If the altitude BD of AABC is intersected by another altitude in G, and EH and HF are perpendicularbisectors, prove BG=2 (HE), and AG 2 (HF). (143.) = H Ex. 289. The diagonals of an isosceles trapezoid are equal. Ex. 290. If the opposite angles of a quadrilateral are equal, the figure is a parallelogram. * Ex. 291. If the diagonals of a trapezoid are equal, the trapezoid is isosceles. Ex. 292. The median drawn to any side of a triangle is less than half the sum of the other two sides. 163. A circumference is a curved line all points of which are equidistant from a point within called the center. A circle is a portion of a plane bounded by a circumference and is usually read ABC or D. A radius is any straight line drawn from the center to the circumference, as DA. A diameter is a straight line passing through the center, and terminating in the circumference; as AE. A semicircum 164. An arc is a part of the circumference. ference is half of the circumference. A minor arc is an arc less than a semicircumference; a major arc is an arc greater than a semicircumference. 165. A secant is a straight line intersecting the circumference in two points; as FG. A tangent is a straight line, which touches the circumference at one point only, and does not intersect it if produced; as MN. 166. A chord is a straight line joining any two points in the circumference; as AB. 167. A central angle is an angle formed by two radii; as A. 168. Circles having the same center are called concentric. PRELIMINARY THEOREMS 169. All radii of the same circle are equal. (By definition.) 170. A point is within, on, or without a circumference, according as the distance from the center is less than, equal to, or greater than a radius. 171. All diameters of the same circle are equal. 172. Two circles are equal if their radii are equal. (Prove by superposition.) 173. A diameter bisects a circle and its circumference. (Prove by superposition.) Ex. 293. The radii of two concentric circles are 6 inches and 9 inches respectively. If a point is 7 inches from the common center, does it lie within the larger circle? Within the smaller? Ex. 294. The distance between the centers of two circles is 4, the radii are 6 and 9 respectively. Does every point of the smaller circle lie within the greater ? PROPOSITION I. THEOREM 174. In the same circle or in equal circles, equal central angles intercept equal arcs; and, conversely, equal arcs subtend equal central angles. HINT. - Prove by superposition. Compare (69). 175. COR. In the same or in equal circles, the greater of two unequal central angles intercepts the greater arc, and conversely. Ex. 295. Divide a circumference into four equal parts. Ex. 297. Divide a circumference into six equal parts. Ex. 298. What is the means for proving the equality of arcs? Ex. 299. If from a point A in a circumference a chord AB and a diameter AC are drawn, a radius parallel to AB bisects arc BC. Ex. 300. If through a point equidistant from two points in the circumference a radius is drawn, the arc between the two points is bisected. Ex. 301. If a secant is parallel to a diameter, the lines intercept equal arcs on the circumference. Ex. 302. Any two parallel secants intercept equal arcs on a circumference. (Ex. 301.) Ex. 303. If the perpendiculars drawn from a point in the circumference upon two radii are equal, the point bisects the arc intercepted by the two radii. Ex. 304. If the line joining the midpoints of two radii is equal to the line joining the midpoints of two other radii, the radii intercept equal arcs respectively. Ex. 305. If the perpendiculars from the center upon two chords are equal, the arcs subtended by the chords are equal. 176. DEF. A polygon is inscribed in a circle, if all its vertices are in the circumference. The circle is then said to be circumscribed about the polygon. PROPOSITION II. THEOREM 177. In the same circle, or equal circles, equal arcs are subtended by equal chords; and, conversely, equal chords subtend equal arcs. |