struction, write out a proof that the figure obtained meets all of the requirements. (5) At the end of the proof discuss the conditions under which more than one figure meeting the requirements may be obtained, or under which no figure meeting the requirements may be obtained. The process suggested in steps (1) and (2) above is called geometric analysis. EXERCISES 1. Inscribe a circle in a given triangle. ANALYSIS. Since O, the required center, must be equidistant from the sides, it must be on the bisectors of A and B, and hence at their intersection. Since AB must be tan gent to the circle, it must be perpen dicular to the radius OF. Hence: CONSTRUCTION. -1. Draw AD and BE bisecting A and 2B, respectively, and let them meet at 0. E C Let the student make the construction and write the construction and proof in full. Do not include the analysis in the written work. 2. Construct three circles each tangent to the other two, with the vertices of a given triangle as x B BC Let the student make the construction and write the complete con struction and proof. 3. Construct a circle which shall be tangent to a given line at a given point and pass through a given external point. ANALYSIS. - If the required circle is tangent to N AB at M, radius OM LAB. If it passes through M and N, MN is a chord, and hence O is on the perpendicular bisector of MN. Hence O is the intersection of the perpendicular to AB at M and the perpendicular bisector of MN. Let the student make the construction, and write the complete construction and proof. 4. In architecture it is sometimes required to draw an easement cornice tangent to the straight cornice at B, and passing through a given point A. Explain the construction and make such a drawing. The same construction is used in laying out the easements of stair rails. must be on a line parallel to AB, at the Let the student make the construction, and write the construction and proof in full. 6. Draw the connecting curve of given radius r at the intersection of the curbs of two given streets. 7. In this base of a column, the lines AC and BD are given, and it is required to connect them by a circular arc with a given radius r. Show how to make the construction, and make a drawing on a large scale of the whole base of a column like this one. 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. equal circles, each tangent to the other two and to the given circle. SUGGESTION. - Construct three equal angles at the center. Then apply Ex. 9. After the first of the three circles is obtained, how may the other two be constructed most easily? 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. F 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. 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. 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. |