(44.) To describe a circle which shall touch a straight line and two circles given in magnitude and position. Let A and B be the centres of the two circles, and CD the line given in position. From B let fall the perpendicular BE, and produce it, making EF= the radius of the circle whose centre is A. Through F draw FG parallel to CD. With the centre B, and radius equal to E F D the difference of the radii of the two circles, describe a circle; through A let a circle be described, touching the line GF and the last described circle (vi. 43.); and let G and H be the points of contact. The centre of this circle will also be the centre of the circle required. Let O be the centre; join OA, OG, OH; and with the centre 0, and radius OI, describe the circle IKL. Since LG=KH=AI, .. OL=OK=OI; the circle IKL .. touches CD in L, and the circle, whose centre is A, in I; and since OB is equal to the difference between OH and HB, i. e. between OA and (IA- BK), or is equal to OK and KB together, .. it touches the circle whose centre is B, in K. (45.) To describe a circle which shall touch two given straight lines, and pass through a given point between them. Let AB, CD be the given lines, and E the given point. Produce the lines to meet in F. Bisect the angle BFD by the line FG; and from E draw EG perpendicular to FG, and produce it both ways to B and D. Take GH=GE; and make DI a mean H proportional between DE and DH; a circle described through the points H, E, I, will touch CD. For the rectangle DE, DH, is equal to the square of DI. And for a similar reason it will touch AB; since the rectangle BH, BE, is equal to the rectangle ED, DH. If the lines AB, CD be parallel ; through the given point E, draw DEHB perpendicular to AB or CD; bisect it in G, and make GH=GE. Take DI a mean proportional between DE and DH; H and a circle described through I, E and H will be the circle required. (46.) To describe a circle which shall touch two given straight lines, and also touch a given circle. Let AB, CD be the given straight lines, EFG the given circle, whose centre is 0. Draw HI, KL parallel to the given lines, so that their perpendicular distances from those lines may be equal to OF the radius of the given circle. By F R B the last problem describe a circle touching HI, KL, and passing through O the centre of the given circle. Let P be the centre of this circle; it will also be the centre of the circle required. Join PM, PN, PO. Since these lines are equal, and MQ, RN, OF are also equal by construction, .. PQ, PR, PF are also equal; and a circle described from the centre P at the distance of any one of them, will pass through the extremities of the other two, and touch the lines AB, CD, in Q and R; since the angles at those points are equal to the angles at M and N, and ... right angles; and it will also touch the circle EFG in F, since OP the line joining the centres passes through F. (47.) To describe a circle which shall touch a circle and straight line, both given in position, and have its centre also in a given straight line. Let the circle whose centre is A, and the straight line BC be given in position; and let CD be the line, in which the centre of the required circle is to be. On BC let fall the perpendicular AB; and make BF F B E H = AE; through F draw FG parallel to BC, meeting DC in G. Join GA; and draw CH parallel to it, meeting the given circle in H, (if the problem be possible). Join AH, and let it meet DC in O. O is the centre of the circle required. Let fall the perpendicular 01. Then (Eucl. vi. 2.) HO OC: AH GC: FB: GC by construction, :: BD: DC, (Eucl. vi. 2.) :: 10: OC, by sim. triangles; .. HO= 10; and a circle described with the centre O, and radius OI or OH, will pass through the extremity of the other, and touch the line BC in I, and the circle in H; because the angles at I are right angles; and AO the line joining the centres of the circles passes through H. (48.) Through two given points within a given circle, to describe a circle, which shall bisect the circumference of the other. Let A and B be the given points within the circle whose centre is 0. Join 40; and produce it indefinitely; and from O draw OC at right angles to it. Join AC; and draw CD at right angles to it, meeting 40 produced in D; and through. A, B, D describe a circle; it will bisect the other in the points E, and F. ..the rectangle AO, OD is equal to the square of OC, i. e. to the rectangle EO, OF; whence (Eucl. iii. 35.) EOF is a straight line; and since it passes through the centre of the circle ECF, it will be a diameter of that circle; the circumference ECF is equal to the circumference EGF, or the circumference of the given circle is bisected. (49.) Through two given points without a given circle, to describe a circle, which shall cut off from the given one an arc equal to a given arc. Let A and B be the given points, and H. CDE the given circle. Join AB; and bisect it in F; from F draw FG at right angles to AB; and from any point G in it, at the distance GA or GB, describe a F circle ABD, cutting the given circle in C and D. Join DC; and produce it to meet BA in H. From H draw HIE (ii. 20.), so that IE may be equal to the chord of the given arc. Through A, B and E describe a circle; it will also pass through I, and cut off the arc required. For the rectangle HI, HE is equal to the rectangle HC, HD, and . also to the rectangle HA, HB, whence I is a point in the circle ABE. (50.) To describe three circles of equal diameters, which shall touch each other. Take any straight line AB, which bisect in D; and from the centres A and B, with the equal radii AD, BD describe two circles. Upon AB describe an equilateral triangle ABC, cutting the circles in E and F; and with the centre C, D B G H E and radius CE or CF, describe another circle; these circles touch each other as required. Since ABAC, and AD is half of AB, .. AE, which is equal to it, is half of AC, and .. AE-EC. In the |