Draw DE D the line which meets the circle, the line which meets the circle shall touch it. Let any point D be taken without the circle ABC, and from it let two straight lines, DCA, DB, be drawn, of which DCA cuts the circle, and DB meets it; and let the rectangle AD, DC be equal to the square on DB. Then DB shall touch the circle. CONSTRUCTION.—Draw the straight line DE, touching the the circle. circle ABC (III. 17); Find F the centre (III. 1' and join FB, FD, FE. PROOF.-Then FED is a right angle (III. 18). And because DE touches the circle ABC, E and DCA cuts it, the rectangle AD, DC is equal to the square on DE (III. 36). But the rectangle AD, DC is equal to the square on DB (Hyp.); Therefore the square on DE is equal to the square on DB (Ax. 1); Therefore the straight line DE is equal to the straight line Then DE = DB. DB. And triangles DEF are And EF is equal to BF (I. Def. 15); Therefore the two sides DE, EF are equal to the two sides DB, BF, each to each; And the base DF is common to the two triangles DEF, DBF and DBF; equal in Therefore the angle DEF is equal to the angle DBF (I. 8). But DEF is a right angle (Const.); spect. Therefore also DBF is a right angle (Ax. 1). And BF, if produced, is a diameter; and the straight line a right angle; which is drawn at right angles to a diameter, from the ex- Therefore DB touches the circle ABC. every re ... DBF is and therefore DB touches the cir:le. EXERCISES ON BOOK III. PROP. 1-15. 1. Two straight lines intersect. Describe a circle passing through the point of intersection and two other points, one in each straight line. 2. If two circles cut each other, any two parallel straight lines drawn through the points of section to cut the circumferences are equal. 3. Show that the centre of a circle may be found by drawing perpendiculars from the middle points of any two chords. 4. Through a given point, which is not the centre, draw the least line to meet the circumference of a given circle, whether the given point be within or without the circle. 5. The sum of the squares of any two chords in a circle, together with four times the sum of the squares of the perpendiculars on them from the centre, is equal to twice the square of the diameter. 6. With a given radius, describe a circle passing through the centre of a given circle and a point in its circumference. 7. If two chords of a circle are given in magnitude and position, describe the circle. 8. Describe a circle which shall touch a given circle in a given point, and shall also touch another given circle. 9. If, from any point in the diameter of a circle, straight lines be drawn to the extremities of a parallel chord, the squares of these lines are together equal to the squares of the segments into which the diameter is divided. 10. If two circles touch each other externally, and parallel diameters be drawn, the straight line joining extremities of these diameters will pass through the point of contact. 11. Draw three circles of given radii touching each other. 12. If a circle of constant radius touch a given circle, it will always touch the same concentric circle. 13. If a chord of constant length be inscribed in a circle, it will always touch the same concentric circle. 14. The locus of the middle points of chords parallel to a given straight line is a line drawn through the centre perpendicular to the parallel chords. PROP. 16-30. 15. Show that the two tangents from an external point are equal in length. 16. Draw a'tangent to a given circle, making a given angle with a given straight line. 17. If a polygon having an even number of sides be inscribed in a circle, the sums of the alternate angles are equal. 18. If such a polygon be described about a circle, the sums of the alternate sides are each equal to half the perimeter of the polygon. 19. If a polygon be inscribed in a circle, the sum of the angles in the segments exterior to the polygon, together with two right angles, is equal to twice as many right angles as the polygon has sides. 20. Draw the common tangents to two given circles. 21. From a given point draw a straight line cutting a given circle, so that the intercepted segment of the line may have a given length. 22. The straight line which joins the extremities of equal arcs towards the same parts are parallel. 23. Any parallelogram described about a circle is equilateral, and any parallelogram inscribed in a circle is rectangular. 24. Two opposite sides of a quadrilateral circumscribing a circle touch the circle at extremities of a diameter. Show that the area of the quadrilateral is equal to one-half the rectangle contained by the diameter, and the sum of the other sides. PROP. 31-37. 25. A tangent is drawn to a circle of 21 inches diameter from a point 17.5 inches from the centre. Find the length of the tangent. 26. Show that a man 6 feet high, standing at the sea level, has a view of 3 miles (approximately) in every direction, along a horizontal plane passing through his eye. 27. The angle between a tangent to a circle and the chord through the point of contact is equal to half the angle which the chord subtends at the centre. 28. From a given point P, within or without a circle, draw a straight line cutting the circle in A and B such that PA shall be three-fourths of PB. Ex. Let the circle be of 1.5 inches radius, and point P 3.5 inches from its centre. Prove your construction by scale. 29. The greatest rectangle which can be inscribed in a circle is a square whose area is equal to half that of the square described upon the diameter as side. 30. If the base and vertical angle of a triangle remain constant in magnitude while the sides vary, show that the locus of the middle point of the base is a circle. 31. Given the vertical angle, the difference of the two sides con. taining it, and the difference of the segments of the base made by a perpendicular from the vertex, to construct the triangle. 32. Show that the locus of the middle point of a straight line, which moves with its extremities upon two straight lines at right angles to each other, is a circle. 33. Show how to produce a straight line, that the rectangle con. tained by the given line, and the whole line thus produced, may be equal to the square of the part produced. Ex. If the length of the given line be 2 inches, show geometrically that the length of the part produced is (V5 + 1) inches. 34. Given the height and chord of a segment of a circle to find the radius of the circle. Ex. If the chord be 24 inches, and the height of the segment be 4 inches, show that the radius of the circle is 20 inches. 35. Show that the locus of the middle points of chords which pass through a fixed point is the circle described as diameter upon the line joining the fixed point and the centre of the given circle. 36. Let ACDB be a semicircle whose diameter is AB, and AD, BC any two chords intersecting in P; prove that ABP = DA AP + CB.BP. MATHEMATICS. SECOND STAGE. SECTION II. ALGEBRA. CHAPTER I. QUADRATIC EQUATIONS. 19, 522 1. Equations of this class, when reduced to a rational integral form, contain the square of the unknown quantity, but no higher powers. When the equation contains the square only of the unknown quantity, and not the first power, it is called a pure quadratic. Thus, aca 25 = 0, 432 + 10 180, are pure quadratics. When the equation contains the square of the unknown quantity, as well as the first power, it is called an adfected quadratic. Thus, 30 = 0, 2c + + 3 = 6, are adfected quadratics. Pure Quadratics. 2. To solve these, we treat them exactly as we do simple equations, until we obtain the value of the square of the unknown quantity; then, taking the square root of each side, we obtain the value of the unknown quantity. It will be 5x = 6, x2 C |