4. OA, OB, OC are three straight lines, and P is any point in OA; show how to draw a straight line through P meeting OB, OC in S, R, so that PS=SR. R Draw PQOC, QR || AO, A 5. Construct a rhombus, having given its diagonals. 6. Construct a square equal to the sum of two given squares. Let AB, CD be sides of the given squares. Draw EF AB, FGLEF and Use Euc. I. 47. = CD. 7. Construct a square equal to the difference of two given squares. E A Let AB, CD be sides of given squares. Draw EF AB, FGLEF; With centre E, and radius equal to CD, 8. Through two points draw two straight lines which shall make with a given straight line a triangle equiangular to a given triangle. 10. Divide a given straight line into any number of equal parts. (A Standard Construction.) Let AB be the given line. Draw AC, and cut off equal parts AC, CD, etc. Join FB, and draw parallels. Then AP=PQ=etc. A R See § 16, Ex. 11. B 11. Divide a triangle into any number of equal parts by lines drawn through the vertex. (A Standard Construction.) Divide the base into the given number of parts as in Ex. 10. Join AD, etc., and apply Euc. I. 38. 12. Construct an isosceles triangle on a given base, having the angle at the vertex double each of the angles at the base. Use Euc. I. 32. 13. Construct an isosceles triangle on a given base, having the vertical angle four times each of the angles at the base. Use Ex. 1. 14. Construct an isosceles triangle on a given base, such that the two angles at the base are together equal to three tinies the vertical angle. 15. Construct an equilateral triangle with a given straight line as altitude. Use Ex. 1. 16. Bisect a parallelogram by a straight line drawn parallel to a given straight line. See § 16, Ex. 16. 17. Draw a straight line through a given point so as to make equal angles with two given straight lines which cannot be produced to meet. Draw parallels through the given point, etc. 18. Divide an equilateral triangle into nine equal triangles whose common vertex is a point within the triangle. See figure of § 13, Ex. 4. 19. Construct a square, having given its diagonal. Compare Ex. 5. 20. Construct a square equal to the sum of three or more given squares. See Ex. 6. 21. Construct a square equal to twice, thrice, four times, etc., the area of a given square. 22. Through two given points draw straight lines which shall make an equilateral triangle with a given straight line. Use the method of Ex. 8. 23. Divide a parallelogram into any number of equal parallelograms by lines drawn parallel to one pair of sides. See Ex. 10, and use Euc. I. 36. 24. Trisect a parallelogram by lines drawn through an angular point. See § 17, Ex. 22. 25. On a given straight line describe a rhombus having an angle equal to a given rectilineal angle. Use the method of Euc. I. 46. 26. On a given straight line describe a rhombus having each of one pair of opposite angles equal to twice each of the other pair. 27. Construct a rectangle equal to a given square, and having one side equal to a given straight line. Use Euc. I. 43, as in I. 44. 28. Trisect a triangle by straight lines drawn from the angular points to meet at a point within the triangle. See § 17, Ex. 6. 29. Draw straight lines through the angular points of a quadrilateral so as to form a parallelogram whose area is double that of the quadrilateral. See § 16, Ex. 17. Show that an infinite number of solutions exist. $23. Loci. An important class of problems is that in which it is required to find a locus. Locus is, of course, nothing else than the Latin word for place, and is the complete answer to a question asking where. For example, Where shall we find a point which is at a given distance AB from a given point C ? If we take a pair of compasses and measure in any direction from C a length CD=AB, D is clearly one answer to the question. By a similar process E is also an answer, and so are F, G, H, etc. In fact there are an infinite number of points which answer the ques tion, and if we continue making dots to represent these points, we shall at last be unable to distinguish the row of dots from a complete circle whose centre is C, and whose radius is equal to AB. This circle is therefore the place where all such points are to be found; and if we suppose a point to start from D and move round the circle, it will at every position satisfy the given condition. The line which it traces out is the gathering together or aggregate of all these positions, and is called the locus of the point. Hence we obtain the following definition : DEFINITION.-The Locus of a point is the aggregate of all the positions of the point which satisfy a given condition. In plane geometry such an aggregate will be found to consist of one or more straight or curved lines or portions of lines. In order to show that any such line (or lines) constitutes the required locus, it is necessary and sufficient to prove (1) that any point which satisfies the given condition is in the line; (2) that every point in the line satisfies the condition. |