Ex. 3. To find on a given circle a point at which a given line shall subtend a given angle. [Discuss the solution as in ex. 1. Show that in certain cases there is no solution unless the given angle is restricted in magnitude.] Ex. 4. To find on a given circle the points at which the angle subtended by a given line is a maximum or minimum. [97, ex. 4.] 197. PROBLEM 22. To find the locus of a point at which the angle subtended by a given circle shall be equal to a given angle. Outline. Show that the locus is a concentric circle, and that its radius may be constructed as follows: Construct a right triangle having one side equal to the radius of the given circle, and the adjacent acute angle equal to the complement of half the given angle; then the hypotenuse is the required radius. Ex. In the four exercises of 196, replace the given line-segment by a given circle, and show that similar solutions can be obtained. INTERSECTION OF LOCI 198. Each of the following constructions is a combination of two locus problems already solved. Ex. 1. To construct a triangle, being given : (a) its base, vertical angle, and altitude (79, I. 255); (c) its base, vertical angle, and sum of squares on sides; Ex. 2. To construct a quadrangle, being given two opposite angles, and three sides; the order in which the five parts are to be taken being specified. Outline. Let the sides AB, BC, CD be given; and also the angles B and D. First construct the triangle ABC (I. 133); and then the triangle ACD (ex. 1, e). [Examine the case in which one of the given angles is convex. Show that there is always a solution when the sum of the two given angles is less than a perigon. Show that there is only one solution.] EXERCISES 1. If from any point on a given circle a line is drawn equal and parallel to a given line, then the locus of the other extremity consists of two circles each equal to the given circle. [Take the given line as "line of translation" (I. 200), and translate the center and any radius, thus reducing the locus problem to a previous one (7).] 2. If from any point on a given circle a line is drawn to a given point, and if this line is turned about the given point through a given angle, then the locus of the other extremity of the line so turned consists of two circles each equal to the given one. [Take the given angle as "angle of rotation" (I. 202), and rotate the circle and any radius about the given point.] 3. To describe three circles of given radii to touch each other externally. 4. To describe three circles of given radii to touch each other so that two may be within the third. 5. In an equilateral triangle the radius of each of the escribed circles is equal to the altitude; and the radii of the circumscribed and inscribed circles are respectively equal to two thirds, and one third of the altitude. 6. If two chords of a circle cut at right angles, then the sum of either pair of opposite arcs is equal to a semicircle. [Through an extremity of one chord, draw a chord parallel to the other, and join its extremity to the other extremity of the first chord.] 7. If two chords of a circle cut at right angles, then the sum of the squares on the four segments is constant, and equivalent to the square on the diameter. 8. If any chord of a given circle passes through a fixed point, then the rectangle of the segments of the chord is constant. 9. If through a fixed point within a given circle any two chords are drawn at right angles, then the sum of the squares on the two chords is constant (exs. 7, 8). 10. If each of two equal circles has its center on the circumference of the other, then the square on their common chord is equivalent to three times the square on the radius. 11. If two given circles have external contact; show how to draw a line through the point of contact so that the whole intercepted part may be equal to a given line. [Form an isosceles triangle whose base is the given line and each of whose other sides equals the sum of the radii. Then the angle which the required line makes with the central line equals one of the base angles; prove.] 12. To construct a triangle, being given the vertical angle, one of adjacent sides, and the perpendicular from the vertex to the base. [Show that the foot of the perpendicular can be found by intersection of loci.] 13. If two circles intersect, then any common tangent subtends, at the common points, angles which are supplemental. 14. If a common tangent is drawn to two circles, and if each point of contact is joined to the two points where the central line meets the corresponding circle, then the two chords so drawn in one circle are respectively parallel to the two chords in the other circle. 15. Find the locus of a point such that the tangent from it to a given circle shall be equal to the line joining it to a given point. [This is a limiting case of the radical axis of two circles (179) when the radius of one of the circles diminishes so that the circle reduces to a point.] 16. Find a point such that the tangent from it to a given circle shall be equal to each of the lines joining it to two given points. 17. Find the locus of the center of a circle which passes through a given point and cuts a given circle orthogonally. [In 188 let one of the circles reduce to a point.] 18. Describe a circle through a given point so as to cut two given circles orthogonally. 19. Describe a circle through two given points so as to cut a given circle orthogonally. 20. Given the vertical angle and the altitude of a triangle, prove that the surface is a minimum when the triangle is isosceles (170). 21. Given the vertical angle and altitude of a triangle, when is the base a minimum ? 22. The inscribed regular hexagon is equivalent to three fourths of the circumscribed one, to half the circumscribed equilateral triangle, and to double the inscribed one, in the same circle. BOOK IV. - RATIO AND PROPORTION 1. That relation between two magnitudes which is expressed by the word ratio will receive a precise definition after certain preliminary notions are explained. The principles will be made sufficiently general to apply to any geometric magnitudes for which appropriate methods of comparison have been given, such as two line-segments, two angles, the surfaces of two polygons, etc. It will not be necessary to restrict our thoughts even to geometric magnitudes. The notion of ratio is applicable to any magnitudes for which the words equivalent, greater, less, sum, difference, etc., have a definite and consistent meaning. Such magnitudes are found in the sciences that deal with number, weight, velocity, probability, etc. MULTIPLES AND MEASURES DEFINITIONS 2. Multiples. If any number of equivalent magnitudes are added together, then their sum is called a multiple of any one of them. If any magnitude P is equivalent to the sum of n magnitudes each equivalent to 4, then P is said to be equivalent to n times the magnitude 4, or to the nth multiple of 4, which is sometimes denoted by the symbol n · A or nA. Thus the double of 4, previously defined, means the same as twice 4, or the second multiple of 4. We may regard A itself as once A, and call it the first multiple of 4. 3. Series of multiples. The magnitudes denoted by the symbols A, 2A, 3A, 4A, NA, ... may be thought of as formed by beginning with the magnitude 4 and successively adding other magnitudes equivalent to 4, as often as desired. The whole set of magnitudes so thought of is called the series of multiples of 4. In considering the mutual relations of certain magnitudes, their series of mutiples will play an important part. 4. Magnitudes of the same kind. Two magnitudes will be said to be of the same kind when their two series of multiples can be directly compared so as to test the equivalence or non-equivalence of any of them. In particular two geometric magnitudes A and B will be said to be of the same kind when it is possible to compare the multiples of 4 with the multiples of B by means of superposition. Such, for instance, are two line-segments, two angles, two equiradial arcs, the surfaces of two polygons; but not a straight line and a curved line, nor two arcs of unequal circles, nor the surfaces of a circle and a polygon. For a simple example of the comparison of two series of multiples, the student may glance forward to the figure in Art. 12, in which the successive multiples of two line-segments A and B are laid off on an indefinite line, all beginning at the same point 0. Again, the natural numbers 1, 2, 3, 4, ... n, **, which we have used to indicate the order in a series of multiples, belong to another class of magnitudes called numerical magnitudes. The terms equivalent (or equal), greater, less, sum, multiple, etc., have definite meanings when applied to them. Thus, the number 3 has its series of multiples, 3, 6, 9, 12, ..., and the number n has its series of multiples, Any two natural numbers are magnitudes of the same kind, since they, or any of their multiples, are directly comparable. |