145. If the bisectors of two angles of an equilateral triangle meet, and, from the point of meeting, lines be drawn parallel to any two sides, these lines will trisect the third side. 146. The bisector of an exterior angle at the vertex of an isosceles triangle is parallel to the base. (122) 147. If, in a triangle ABC, the bisectors of an interior angle at B and of an exterior angle at C meet in D, then D = ZA. 148. The angle formed by the bisectors of any two consecutive angles of a quadrilateral is equal to the sum of the other two angles. (125) 149. The sum of the distances of any point in the base of an isosceles triangle from the arms, is constant; i.e., is always equal to the altitude upon an arm. (143) 150. The sum of the distances of any point within an equilateral triangle from the sides is constant; i.e., is equal to an altitude. (Exercise 148) 151. The lines joining the mid points of the sides of a triangle divide it into four equal triangles. (152) 152. The three altitudes of a triangle have a common point. (141, 155) 153. The lines joining the mid points of adjacent sides of any quadrilateral, form a parallelogram the sum of whose sides is equal to that of the diagonals of the quadrilateral. (147) 154. The three medians of a triangle have a common point which cuts off one third of each median. (147) (120) B D 155. If the exterior angles of a triangle are bisected, the three exterior triangles formed on the sides of the original triangle are equiangular. 156. The vertices of all right triangles having a common base as hypotenuse, lie in the same circumference. (Exercise 144) BOOK II. THE CIRCLE. LOCI. PROBLEMS. ELEMENTARY PROPERTIES. 158. A circle is a plane figure bounded by a curved line such that a certain point within, called the center, is equidistant from every point of the curve. A 159. The circumference of a circle is 4 the curve that bounds it; as ADEBC. 160. An arc is any part of a circumference; as AD or ACE. B E 161. A radius is any straight line drawn from center to circumference; as OA. 162. COR. All radii of the same circle are equal (158). Also, a point is within, on, or without a circumference according as its distance from the center is less than, equal to, or greater than a radius. 163. A chord is a straight line joining the extremities of an arc; as DE. The chord is said to subtend its arc. 164. A diameter is a chord that passes through the center; as AB. 165. COR. All diameters of the same circle are equal. For each is double the radius of that circle. PROPOSITION I. THEOREM. 166. The straight line terminating in any two points in a circumference lies wholly within the circle. Given: A straight line, AB, terminating in two points in the circumference whose center is 0; To Prove: AB lies within that circumference. Take any point P between A and B, and join OA, OB, OP. i.e., P, which is any point between A and B, lies within the circle. Q.E.D. 167. COR. A straight line can meet a circumference in not more than two points. EXERCISE 157. Show that a circle cannot have more than one center. State the axioms upon which your proof depends. 158. A straight line will cut a circle, or lie entirely without it, according as its distance from the center is less than, or greater than, the radius of the circle. 159. If a straight line could meet a circumference in three points, how many equal straight lines could be drawn from the center to that PROPOSITION II. THEOREM. 168. Circles having equal radii are equal. ӨӨ D Given: Two circles, ABD, A'B'D', having radius 04 equal to radius O'A'; To Prove: Circle ABD is equal to circle A'B'D'. Place ABD upon O A'B'D', so that OAO'A'. Then, not only will A coincide with A', but also circumference ABD with circumference A'B'D', since all points of both circumferences are equally distant from the coinciding centers 0 and o' (Hyp. and 158). Hence, since circumf. ABD circumf. A'B'D', O ABDO A'B'D'. Q.E.D. (14) 169. COR. A diameter bisects the circle and its circumference. For producing 40, 4'0', to form the diameters AC, A'c'; since the part ABC may be made to coincide, as above, either with A'B'C' or A'D'C', these parts are equal, as are also their bounding arcs. (Ax. 1) 170. DEFINITION. A half circle is called a semicircle; a half circumference, a semicircumference. EXERCISE 160. How many points are necessary to determine the magnitude and position of a circle in a given plane? 161. Can two circles have a common center without coinciding? CHORDS. PROPOSITION III. THEOREM. 171. If a chord is perpendicular to another chord at its mid point, the first chord is a diameter. E B Given: Chord AB perpendicular to chord CD at its mid point E; To Prove: AB is a diameter. 172. COR. A radius perpendicular to a chord bisects that chord. (97) EXERCISE 162. In the diagram for Prop. III., if AC, AD, be joined, then AC AD. = 163. The line drawn through the mid points of parallel chords in a circle, passes through the center. 164. If an isosceles triangle be constructed on any chord of a circle, its vertex will be in a diameter or a diameter produced. 165. If two chords of a circle bisect each other, both are diameters. 166. Enunciate the converse of Prop. III. Is that converse always true? 167. If from any point within a circle two equal straight lines be drawn to the circumference, the bisector of the angle they form will pass through the center. Geom. - 6 |