arcs are to be tangent to each other, they must be tangent to what lines? Since each arc is to be tangent to both diagonals, its center must be on what line? 9. Construct a circle tangent to a given arc and to the sides of the central angle which intercepts the arc. ANALYSIS. - To what point of AB is radius CD drawn? How is EF drawn? How are points E and F located? Then the required circle is inscribed in AEFC. See Ex. 1. Make the construction, and write the complete construction and proof.. 10. In a given circle construct three C B E D F equal circles, each tangent to the other two and to the given circle. 11. The construction of a trefoil, formed by the arcs of three equal circles in a given circle as shown in the figure, is encountered frequently in architectural designs. Explain the construction, and draw a trefoil in a given circle. SUGGESTION. - Use a circle at least three inches in diameter. Proceed as in Ex. 10. 12. Show how to locate the centers of the circular arcs in this rosette. Draw a rosette like this three inches in diameter. SUGGESTION. - Proceed as in Ex. 10. 13. This tracery window contains three equal circles within an equilateral triangle, each circle being tangent to the other two circles and to the two sides of the triangle. Construct three such circles within a given triangle. SUGGESTION. - Since the circles are tangent to each other, they must have a common tangent line at each point of contact. Then each circle is inscribed in a triangle how found? 14. Construct within a given square four equal circles, each tangent to two others and to two sides of the square. 15. Construct a circle with given radius which shall pass through two given points. 16. Construct a circle with given radius tangent to a given circle and to a given straight line. 17. Construct a circle with given radius which shall be tangent to two given circles. 18. Construct a circle which shall pass through a given point and be tangent to a given circle at a given point. 19. Construct a circle with given radius which shall pass through a given point and be tangent to a given circle. 20. Through a given point draw a straight line cutting a given circle so that the chord intercepted on it by the circle shall equal a given length. 21. In a given circle draw a chord of given length and parallel to a given straight line. 22. Through an intersection of two given circles draw a line-segment of given length terminating in the two circles. 23. Draw a tangent to a given circle which shall be perpendicular to a given straight line. 24. Draw a tangent to a given circle which shall be parallel to a given straight line. Construct a right triangle, having given: 25. One leg and the hypotenuse. 26. One leg and the altitude to the hypote nuse. ANALYSIS. - If & ABC is the required triangle, with leg AB equal to the given segment a, and AD, altitude to hypotenuse BC, equal A to the given segment b, then BC is tangent to a circle with center A and radius AD. a b A! B Make the construction, and write out the complete construction and proof. 27. The hypotenuse and the altitude to the hypotenuse. 28. An acute angle and the altitude to the hypotenuse. 29. An acute angle and the sum of the legs. ANALYSIS. - If ∠ x is the given angle, AB the sum of the legs, and DBC the required triangle, then CD=AD. Hence ∠DAC = 45°. Therefore begin by constructing an angle of 45° at A. Make the construction, and write the complete construction and proof. 30. The hypotenuse and the sum of the legs. A C 31. Construct an equilateral triangle, having given an altitude. B 32. Construct an isosceles triangle, having given the base and the angle at the vertex. 33. Construct a triangle, having given two sides and the angle opposite one of them. 34. Construct a triangle, having given a side, the angle opposite it, and the altitude to another side. 35. Construct a triangle, having given a side, the median to that side, and the altitude to that side.. 36. Construct a triangle, having given two sides and the altitude to one of the given sides. 37. Construct a triangle, having given a side and the altitudes to the other two sides. 38. Construct a triangle, having given a side, the altitude to the given side, and the radius of the circumscribed circle. 39. Construct a triangle, having given one side, an adjacent angle, and the sum of the other two sides. 40. Construct a triangle, having given the three medians. 41. Construct an isosceles triangle, having given the perimeter and the altitude to the base. ANALYSIS. - If a is the given altitude, AB the given perimeter, and DEF the required triangle, then DE = AE and DF = BF. How, therefore, are E and F located? Make the construction and write the construction and proof in full. A ID E CF 山 B 42. Draw a tangent to a given circle at a given point when the center is inaccessible. 43. Draw the bisector of a given angle when the vertex is inaccessible. CHAPTER IX NUMERICAL RELATIONS 187. Numerical relations. - The numerical measure of a geometric figure was defined in § 115. A proportion between the parts of similar figures was defined in Chapter VI as a certain relation between their numerical measures. In the present chapter certain proportions and certain other numerical relations of geometric figures that are derived from proportions will be considered. Since the terms of a proportion between the parts of geometric figures are taken as abstract numbers, they may be multiplied or subjected to any other of the operations of abstract numbers. Thus, in the sections which follow, the product of two line-segments AB and CD shall be taken to mean the product of the abstract numbers corresponding to their numerical measures. 2. A median to a side of a triangle bisects any line-segment parallel to that side and terminating in the other two sides. |