tant from A and C and four miles from B. Is there only one? 67. What is the locus of the vertex of an isosceles A having a given base? Each locus requires a geometrical proof; e. g., in Ex. 62—to prove the answer, "The locus of a point equidistant from two given points is the right bisector of a line joining the given points," fix the points, draw the line joining them, and the right bisector. Then select any point in the locus (the right bisector) and prove that it is equidistant from the given points. Then select any point without the locus and prove that it is not equidistant from the points. In general we may say that when we prove a theorem concerning the locus of points, it is necessary to prove two things: (1) That all points in the locus satisfy the given conditions. (2) That any point not in the locus does not satisfy the given conditions. Can you tell why both of the above proofs are necessary? BOOK II. CIRCLES. Define (1) a circle, (2) a circumference, (3) an arc, (4) a chord, (5) a radius, (6) a diameter. Are all radii and all diameters of the same or equal circles equal? 127. A segment of a circle is a part cut off by a chord. bounded by an arc and the chord cutting it off. 128. It is A semi-circle is one-half of a circle. A semi-circumference is one-half of the circumference. A semi-circle is bounded by a diameter and a semi-circumference. 129. A sector is a part of a circle bounded by an arc and two radii. 130. A tangent to a circle is a line which touches it, but will not enter the circle, no matter how far the tangent is produced. The point of tangency is the point where the line touches the circle. Two circles are tangent when their circumferences touch but do not cut each other. They may be tangent internally when one is wholly within the other; or tangent externally when one is wholly without the other. 131. A secant is a line which cuts the circumference in two points. 132. A central angle is an angle formed by two radii. 133. (1) An inscribed angle is an angle whose vertex is a point in the circumference, and whose sides are chords. (2) An inscribed polygon has its vertices in the circumference of the circle. 134. PROPOSITION I. Which chord divides the circle into two equal parts? Prove your answer. State Prop. I. 135. PROPOSITION II. Draw a circle and any chord not the diameter. Compare the parts of the chord. State and prove Prop. II. 136. Cor. I. Draw a circle and a chord. Draw the right bisector of th chord. Will it pass through the center of the circle? State and prove Cor. I., Prop II. [Hint.—If you fail to prove the above, consult § 111.] 68. EXERCISE. Construct a chord when the circle and the mid-point of the chord are given. (See "hint" below if you fail.) [Hint.-Draw the radius through the given point; then draw a 1 to the radius at that point.] 137. PROPOSITION III. (1) Draw in equal circles two equal central angles. Do the arcs which subtend the equal angles appear to be equal? [How did you prove that circles having equal radii or equal diameters are equal? How then can you prove arcs equal?] (Subtend means to be below, or under, or to be opposite; e.g., arcs are said to subtend central angles, and chords subtend arcs.) (2) Again, in equal circles take equal arcs. Do the central angles appear equal? State the truths discovered as one proposition, the second part being the converse of the first. Prove Prop. III. 138. Cor. I. Draw a circle and a chord. Draw the right bisector of the chord. Does it pass through the center? [Auth.] Prove your answer. Does it bisect the subtended arc? Prove it. State the truths discovered as one corollary. EXERCISE. 69. (1) Bisect a given arc. (2) Bisect a given angle, not as in § 46. 139. PROPOSITION IV. Given two equal circles and two equal arcs. Compare the chords which subtend the arcs. State and prove Prop. IV. EXERCISE. 70. Construct and prove § 49 again, using a method you were unable to use when first given. |