Then every circle which has its centre in AE, and passes through P, must also pass through P'. Hence the problem is now reduced to drawing a P and P' to touch either AC or AB. Ex. 1, p. 233. circle through Ex. 21, p. 253. Produce P'P to meet AC at S. Describe a square equal to the rect. SP, SP'; and cut off SR equal to a side of the square. Describe a circle through the points P', P, R. Then since the rect. SP, SP' the sq. on SR, this circle touches AC at R; II. 14. Constr. and since its centre is in AE, the bisector of the L BAC, III. 37, Q.E.F. NOTES. (i) Since SR may be taken on either side of S, it follows that there will be two solutions of the problem. (ii) If the given straight lines are parallel, the centre lies on the parallel straight line mid-way between them, and the construction proceeds as before. 24. To describe a circle to touch two given straight lines and a given circle. Let AB, AC be the two given st. lines, and D the centre of the given circle. It is required to describe a circle to touch AB, AC and the circle whose centre is D. Draw EF, GH par1 to AB and AC respectively, on the sides remote from D, and at distances from them equal to the radius of the given circle. Describe the pass through D. MND to touch EF and GH at M and N, and to Let O be the centre of this circle. Ex. 23, p. 254. Join OM, ON, OD meeting AB, AC, and the given circle at P, Q, and R. Then a circle described with centre O and radius OP will touch AB, AC and the given circle. For since O is the centre of the MND, .. OM ON=OD. But PM QN=RD; Constr. a circle described with centre O, and radius OP, will pass through Q and R. And since the Ls at M and N are rt. angles, .. the 4s at P and Q are rt. angles; the PQR touches AB and AC. III. 18. I. 29. And since R, the point in which the circles meet, is on the line of centres OD, the PQR touches the given circle. Q.E.F. NOTE. There will be two solutions of this problem, since two circles may be drawn to tcuch EF, GH and to pass through D, 25. To describe a circle to pass through a given point and touch a given straight line and a given circle. Let P be the given point, AB the given st. line, and DHE the given circle, of which C is the centre. It is required to describe a circle to pass through P, and to touch AB and the DHE. Through C draw DCEF perp. to AB, cutting the circle at the points D and E, of which E is between C and AB. Join DP; and by describing a circle through G F, E, and P, find a point K in DP (or DP produced) such that the rect. DE, DF = the rect. DK, DP. Describe a circle to pass through P, K, and touch AB: Ex. 21, p. 253. This circle shall also touch the given O DHE. For let G be the point at which this circle touches AB. Join DG, cutting the given circle DHE at H. Join HE. Then the DHE is a rt. angle, being in a semicircle, also the angle at F is a rt. angle; ..the points E, F, G, H are concyclic: the rect. DE, DF = the rect. DH, DG: but the rect. DE, DF = the rect. DK, DP: the rect. DH, DG= the rect. DK, DP: ..the point H is on the OPKG. Let O be the centre of the PHG. Join OG, OH, CH. III. 31. III. 36. Constr. Then OG and DF are par1, since they are both perp. to AB; and DG meets them. ..the OGD = the L GDC. But since OG=OH, and CD=CH, I. 29. the OGH = the L OHG; and the ▲ CDH the 4 CHD: NOTES. (i) Since two circles may be drawn to pass through P, K and to touch AB, it follows that there will be two solutions of the present problem. (ii) Two more solutions may be obtained by joining PE, and proceeding as before. The student should examine the nature of the contact between the circles in each case. 26. Describe a circle to pass through a given point, to touch a given straight line, and to have its centre on another given straight line. 27. Describe a circle to pass through a given point, to touch a given circle, and to have its centre on a given straight line. 28. Describe a circle to pass through two given points, and to intercept an arc of given length on a given circle. 29. Describe a circle to touch a given circle and a given straight line at a given point. 30. Describe a circle to touch two given circles and a given straight line. V. ON MAXIMA AND MINIMA. We gather from the Theory of Loci that the position of an angle, line or figure is capable under suitable conditions of gradual change; and it is usually found that change of position involves a corresponding and gradual change of magnitude. Under these circumstances we may be required to note if any situations exist at which the magnitude in question, after increasing, begins to decrease; or after decreasing, to increase in such situations the magnitude is said to have reached a Maximum or a Minimum value; for in the former case it is greater, and in the latter case less than in adjacent situations on either side. In the geometry of the circle and straight line we only meet with such cases of continuous change as admit of one transition from an increasing to a decreasing state-or vice versâ-so that in all the problems with which we have to deal (where a single circle is involved) there can be only one Maximum and one Minimum-the Maximum being the greatest, and the Minimum being the least value that the variable magnitude is capable of taking. Thus a variable geometrical magnitude reaches its maximum or minimum value at a turning point, towards which the magnitude may mount or descend from either side: it is natural therefore to expect a maximum or minimum value to occur when, in the course of its change, the magnitude assumes a symmetrical form or position; and this is usually found to be the case. This general connection between a symmetrical form or position and a maximum or minimum value is not exact enough to constitute a proof in any particular problem; but by means of it a situation is suggested, which on further examination may be shewn to give the maximum or minimum value sought for. For example, suppose it is required to determine the greatest straight line that may be drawn perpendicular to the chord of a segment of a circle and intercepted between the chord and the arc : we immediately anticipate that the greatest perpendicular is that which occupies a symmetrical position in the figure, namely the perpendicular which passes through the middle point of the chord; and on further examination this may be proved to be the case by means of I. 19, and I. 34. Again we are able to find at what point a geometrical magnitude, varying under certain conditions, assumes its Maximum or Minimum value, if we can discover a construction for drawing the magnitude so that it may have an assigned value: for we may then examine between what limits the assigned value must lie in order that the construction may be possible; and the higher or lower limit will give the Maximum or Minimum sought for. It was pointed out in the chapter on the Intersection of Loci, [see page 125] that if under certain conditions existing among the data, two solutions of a problem are possible, and under other conditions, no solution exists, there will always be some intermediate condition under which one and only one distinct solution is possible. Under these circumstances this single or limiting solution will always be found to correspond to the maximum or minimum value of the magnitude to be constructed. 1. For example, suppose it is required to divide a given straight line so that the rectangle contained by the two segments may be a maximum. We may first attempt to divide the given straight line so that the rectangle contained by its segments may have a given area—that is, be equal to the square on a given straight line. Let AB be the given straight line, and K the side of the given square. X A M' Z P Z M B It is required to divide the st. line AB at a point M, so that the rect. AM, MB may be equal to the sq. on K. Adopting a construction suggested by II. 14, describe a semicircle on AB; and at any point X in AB, or AB produced, draw XY perp. to AB, and equal to K. Through Y draw YZ par1 to AB, to meet the arc of the semicircle at P. Then if the perp. PM is drawn to AB, it may be shewn after the manner of II. 14, or by III. 35 that the rect. AM, MB the sq. on PM So that the rectangle AM, MB increases as K increases. Now if K is less than the radius CD, then YZ will meet the arc of the semicircle in two points P, P'; and it follows that AB may be divided at two points, so that the rectangle contained by its segments may be equal to the square on K. If K increases, the st. line YZ will recede from AB, and the points of intersection P, P' will continually approach one another; until, when K is equal to the radius CD, the st. line YZ (now in the position Y'Z') will meet the arc in two coincident points, that is, will touch the semicircle at D; and there will be only one solution of the problem. If K is greater than CD, the straight line YZ will not meet the semicircle, and the problem is impossible. Hence the greatest length that K may have, in order that the construction may be possible, is the radius CD. .. the rect. AM, MB is a maximum, when it is equal to the square on CD; that is, when PM coincides with CD, and consequently when M is the middle point of AB. NOTE. The special feature to be noticed in this problem is that the maximum is found at the transitional point between two solutions and no solution; that is, when the two solutions coincide and become identical. |