T A K к D R But the circle NPQ stands related to each of the circles ABC, DEF, with the corresponding points M, L, in the same manner in which KPQ is related to BCD in the preceding locus. For, with regard to ABC, because GB is perpendicular to BR, which is the diameter of the semicircle BLMR, GB touches the semicircle (2.) at B, and therefore (21.) GB2 is equal to GLX Secondly, let it be "required to find GM, that is, GL a third proportional to the locus of all points P such that the GM and GB, or GC; and KN was tangent PA, drawn to a given circle made a mean proportional between ABC, and the tangent PD drawn to a KL and KM. And, in the same mansecond given circle DEF, may be to one ner, with respect to the other circle another in a given ratio. DEF, HM is a third proportional to Take G, H the centres of the two cir- HL and HF; and KN was made a cles; join G H, and produce it on the KM. Therefore, by the last question, mean proportional between KL and PA2 is to PM as KG to KM; and PL is to PDR as KL to KH. Also, because KL, KN and KM are proportionals, PM® MHK is to PL' as KM to KL. (51.) Therefore the ratio of PA? to PDR is compounded of ratios which are the same with the ratios of KG to KM, KM to KL, and KL to KH, that is, it is the same side of the lesser circle to O so that OG with the ratio of KG to KH (II. 27.); may be to OH as the radius GC to the and PA is to PD in the subduplicate radius HF (II. 55.) : upon Og describe ratio of KG to KH, that is in the given a semicircle cutting the circle ABC in ratio. Therefore the circle NPQ is the B, so that OB and BG being joined locus required. may be perpendicular (15. Cor. 1.) to one If the given ratio be the ratio of another, and therefore OB a tangent at equality, the difference of the squares B (2.); and from H to OB (produced if necessary) draw the perpendicular HR. Then, because HR is parallel to GB (I. 14.) HR is to GB as OH to OG (11.30.Cor.2.), that is, as HF toGC; and, because GB is equal to GC, HR is equal to HF. Therefore R is a point in the circle DEF; and because ORH is a right angle, OB touches the circle DEF in R. Upon BR describe the semicircle of PG, PH will be equal to the differBLMR cutting GH in the points L and ence of the squares of GC and HF; M; and if PA is to be greater than and therefore the locus is a straight line PD, produce GH to K (II. 55.) so that (49.) cutting GH at right angles, and GK may be to HK in the duplicate of may be determined as in Prop. 49. the given ratio (as in the figure); but, The first of the two loci we have thus if PĂ is to be less than PD, produce discussed is manifestly the same which HG to K (II. 55.), so that GK may be satisfies the condition that A and E to HK in the duplicate of the given being two given points, and ED a ratio; take KN (II. 51.) a mean propor- given square, PAO shall be to PEPtional between KL and KM, and from EDP in a given ratio : and the second, the centre K with the radius KN describe the same which satisfies the condition the circle NPQ; this circle shall be that G and H being two given points, the locus required. and GC2 and HF2, two given squares, For, if P be any point in the locus, PGP-GC shall be to PH:-HF in a and if the tangents PA and PD be given ratio. drawn, and PL, PM joined, PA will be to PD in the ratio which is com PROP. 52. pounded of the ratios of PAP to PM, A point A being given within oja PM to PL, and PLto PD?.(11. def.12.) without a circle BDI, and in every А. D B chord M N which passes through it, a passes through it, tangents being drawn second point P being taken such that the intersecting in P; it is required to find chord produced may be divided by these the locus of the points P. two points and the circumference har Let C be the centre of the circle, and monically; it is required to find the let C A, produced if necessary, meet the locus of the points P. circumference in B: take CF a third Let C be the centre of the circle, and proportional to C A,C B: join PF, PC, let C A, produced if necessary, meet the CM, CN, and let PC cut M N in Q. circle in B: take CF a third propor Then, because PM is equal to PN tional to CA, CB, (11. 52.) and join FP, (2. Cor. 3.), and CM to CN, MN is biFM, FN. Then, because CF, CB, CA sected by PC at right angles (3. Cor. 3.). are proportionals, and that CE is equal And, because CNP is a right angled to CB, the straight lines EA, EB, EF triangle (2.), and that from the right are in harmonical progression (II. 46.). angle N, a perpendicular N Q is drawn And, because upon the mean EB, the to the hypotenuse, the rectangle under circle EDB is described (51. Cor.), AM CQ, CP is equal (I. 36. Cor. 2.) to the is to M Fas AN to NF. Therefore, alternando,(II. 19.) A M:AN::MF: NF. But by the supposition that P is a point of the required locus, MP:PN:: AM : AN, that is, :: MF:NF: therefore, in the triangle FMN, both the base M N and the base produced are divided in the ratio of the sides. F D P square of CN, that is, to the square of CB, or to the rectangle CA, CF (II. 38. Cor. 1.). Therefore (II. 38.) CQ is to CA as CF to CP, and (II. 32.) the triangle CFP is equiConsequently, as was "shown in the angular with the triangle C Q A. There shown in the fore CFP is a right angle, and the like case in the demonstration of the last point P is in a straight line drawn gle , and the point P lies in a straight line through the point F perpendicular to drawn from the point F perpendicular to and to show that every point in this CF. It is easy to reverse the reasoning, CF. It is easy to reverse the reasoning, straight line satisfies the given condition. and to show that every point in this Therefore this straight line is the locus straight line satisfies the given condition. required. Therefore this straight line is the locus Cor. If the diameter of a circle, and required. Cor. If the diameter of a circle, and the diameter produced, be divided in the the diameter produced, be divided in the (II. 45 Cor.) if the diameter produced be same ratio, or, which is the same thing, same ratio, or, which is the same thing, divided harmonically, and if tangents be (11.45 Cor.) if the diameter produced be drawn at the extremities of any chord divided harmonically, any chord which passes through one point of division shall passing through one of the points of dibe divided harmonically by the circum- vision, they shall intersect one another ference and the perpendicular to the which is drawn through the other point. in the perpendicular to the diameter diameter which is drawn through the other point. SECTION 7.–Problems. PROP. 54. Prob. 1. (Euc. iii. 30.) A point A being_given within or To bisect a given circular arc ACB. without a circle BD E, and at the ex Let C be the required point of bitremities of every chord M N which section: take D. the middle point of A D B B AB, and join CA, CB, CD. Then, Since the straight because the arc CA line which bisects a is_equal to the arc chord at right angles CB, the chord CA (12. passes through the Cor. 1.) is equal to the centre of the circle, chord C B: and, be two such straight lines cause the triangles CDA, CDB have will cut one another in the centre. the three sides of the one equal to the Therefore, in the arc A CB take any three sides of the other, each to each, point C; join A C, C B; and bisect AC, the angle CDA is equal to CDB: there. C B at right angles by the straight lines fore, the line CD bisects AB at right DE, FE: the point E in which they angles. Therefore reversely, bisect the cut one another is the centre of the arc chord AB at right angles by the straight A CB. (See also Prop. 44.) line CD, and the given arc ACB will be Cor. (Euc. iïi. 25.) Hence, any arc of bisected in C. a circle being given, the circumference Therefore, &c. may be completed of which it is a part. Cor. Hence, a given arc may be divided into 4, 8, 16, &c. equal parts. PROP. 56. Prob. 3. (Euc. ii. 17.) Scholium. From a given point A, to draw tanThe practicability of a geometrical gent to a given circle B DE. division of a circular arc into any num- 1. If the point A ber of equal parts, implies that of the be in the circumfeangle at the centre (12.) into the rence of the circle, same number of equal parts; and vice find the centre Ć D versâ. It has already been stated that (55.), join CA, in the cases of 3, 5, &c., equal parts, and from A draw the division of the angle cannot be ef- A F perpendicular fected by a plane construction; and the to C A. "Then, because AF is drawn same is to be understood of the circular perpendicular to the radius at its extrearc (I. 46. Scholium). We may observe mity, it touches the circle (2.). that the problem of trisecting an arc 2. If A do not lie has been put under the following form, in the circumference, which gives it an appearance at first of let the line A B be being much easier than upon examina- assumed as the retion it is found to be. quired tangent. Find “ From a given the centre C, and join point A in the cir. CA, CB. Then, becumference of cause AB is a tangent, the angle CBA is a given circle ABD B right angle (2. Cor. 1.), and the point B to draw a straight lies in the circumference of a circle of line APQ such that which AC is diameter (15. Cor. 3.) the part PQ, which Reversely, therefore, upon A C as a diais intercepted between the circumference meter describe a circle cutting the given and a given diameter BD produced, shall circle in B, and join AB: A B is the be equal to the radius CA." tangent required. For, if this be done, the arc PD will We may observe that in this case be found, which is a third of the given there are two points of intersection B, arc AB; because, PQ being equal to and therefore two tangents. The same PC, the angle PCQ is equal (1.6) to the may be said, indeed, of the former case; angle PQČ, and therefore, the angle but there the two touch in the same CPA or CAP is equal to twice PCQ point, and are parts of the same straight (I. 19.), and ACB, which (I. 19.) is equal line. to CAP and PQC together, is equal to three times PCQ, that is, the arc PROP. 57. Prob. 4. AB is equal to three times the arc PD (13.) To a given circle A B D to draw a tangent, which shall be parallel to a PROP. 55. Prob. 2. (Euc. iii. 1.) given straight line E F. To find the centre of a given circular Find the centre C (55.); from C draw arc A C В. . CE perpendicular to EF (I. 45.), and D 33 а P. B P E E B В B let CE, produced if upon opposite sides of the line joining necessary, cut the cir the centres, the like reasoning will lead cumference in to a construction which differs from Then, if PQ be drawn that of the first case in this only, that parallel to EF, it will CE is equal to the sum of the radii. be at right angles to We may remark that in each of the the radius CP (I. cases there are two tangents c E, and 14.); and therefore two common tangents parallel to them (2.) will be the tan respectively. gent required. PROP. 59. Prob. 6. To describe a circle, which shall touch each of two given circles A BD, 1. (Euc. iv. 5.) pass through three a bd. given points not in the same straight Let A a be assumed as the common line; or tangent required: and first, let it touch 2. pass through two given points, the circles in the points A, a which lie and touch a given straight line ; or upon the same side of the line joining touch two given straight lines ; or 3. pass through a given point, and their centres C, c. 4. (Euc. iv. 4.) touch three given straight lines not parallels. 1. Let A, B, C be the three given points, and let the point P be assumed for the centre of the required circle. Then, because P is equidistant from A and B, it is in the Join C A,ca; and from c, the centre straight line which of the lesser circle, draw cE parallel bisects A B at right to A a to meet CA in E. Then, be- angles (44.); and cause C Aa (2. Cor. 1.) is a right angle, for a similar reason, CEc is likewise a right angle (I. 14.), it is in the straight and C E will touch in the point E the line which bisects BC at right angles. circumference of a circle, described Therefore, reversely, to find the point P, from the centre C, with the radius CE, bisect AB, BC at right angles (1.43. Cor.) which is equal to the difference of CA, by the straight lines DP, EP, which inEA, that is (I. 22.) to the difference of tersect one another in P: and, from the the radii CA, C a. centre P with the radius PA, describe Therefore, reversely, from the centre C with a radius equal to the difference B, C, and shall be the circle required. a circle; it shall pass through the points of the radii CA, ca, describe a circle, 2. Let A, B be the two given points, and from the point c draw a straight and CD the given straight line. Supline (56.), touching this circle in the pose the circle to be described, and that point E : join CE, and produce it to meet it touches C D in P: also, let A B the circumference ABD in A; draw c a parallel to C A, and join A a: 'Aa is the produced meet CD in C. Then, becommon tangent required. When the circles are equal, CA and camust be drawn, each of them, perpendicular to Cc. In this case the common tangent A a is evidently. parallel to C c. 2. If the points of contact are to lie cause C P touches, and C A cuts the circle, the square of C P is equal (21.) to the rectangle under CA, C B. Therefore, reversely, take C Pa mean proportional between CA, CB, and describe a circle through the points A, B, P; it shall be the circle required. We may remark that in this case, there may be described two circles, A B D P Р D B В D D B B each of which passes through the two In both cases we may observe that given points A, B, and touches the there are two circles which satisfy the given straight line CD, viz. one for a given conditions, corresponding to the point P upon the right hand of C, and two points in which the circle which is another for a point P upon the left described from the centre G with the rahand of C. dius G E cuts the line FA. If AB be parallel to CD, the point 4. Let A B, BC, P will be in the straight line which bi- CD be the three sects A B at right angles, (3. Cor. 1.) given straight lines, and being found accordingly, the circle of which not more may be described as before. than two are paral .A 3. Let A be the given point, and BC, lel, and let these DE the given straight lines. Produce two be cut by the BC, DE to meet one another in F: third in the points join FA: draw F G bisecting the angle B,C. BFD (1.46.): in FG take any point Assume the point P for the centre of G, and from G draw G E (1. 45.) per- the circle. Then, because P is equipendicular to FD: from the centre G distant from AB, BC, it is in the with the radius GE describe a circle straight line B P which bisects the ancutting FA in H: join HG, and gle ABC (45.); and, for a similar reason, through A draw A P parallel to HG it is in the line PC which bisects the to meet FG (produced, if necessary), in angle BCD. Therefore, reversely, to P (I. 48.): the circle described 'from find P, bisect the angles at B and C by straight lines meeting in P: from Ý draw P Q perpendicular to A B (I. 45.), and from the centre P with the radius P Q describe a circle: it shall be the circle required. If AB, C D are paralle), two circles (and two only) can be described, each touching the three given lines: but if no two of the straight lines be paralthe centre P with the radius P A shall lel, four circles may be described w} shall satisfy be the circle required. this condition, viz. one For, if PD be drawn from the point given lines, and three others touching within the triangle included by the P perpendicular to FD, PD will be the sides of that triangle externally. parallel to G E (I. 14.), and, therefore, (II. 30. Cor. 2.) GE will be to PD as GF Scholium. to PF: but, because G H is parallel to PA, G F is to PF as GH to PA: The problem of describing a circle therefore (II. 12.) G E is to PD as GH about a given triangle (Euc. iv. 5.) beto PA: and in this proportion the first longs to the first case, that of inscribterm GE is equal to the third GH; ing a circle within a given triangle therefore also (II. 18.) PD is equal to (Euc. iv. 4.) to the last case of this proPA. Therefore, the circle which is de- position. The second and third cases scribed from the centre P with the ra are modified by supposing a point and a dius P A passes through the point D; tangent passing through it to be of the and it touches the line D E in that data. Thus, the second becomes “ to point, because PD is at right angles to describe. a circle, which shall pass DE (2.): therefore (45.) it also touches through two given points, and touch a the line B C.* given straight line in one of those The case in which B C is parallel to points," and the third “to describe a DE differs from the foregoing in this circle which shall touch two given only, that FG must now be drawn straight lines, and one of them in a parallel to BC or DE, and bisecting given point." The modified solutions the distance between them (See 45.). corresponding are too simple to detain us here: that of the first occurs in prob. 7. * When the point A is in FG, or in FG proluced, Instead of touching one, two, or three the solution here given must be modified by joining HE, drawing A D parallel to H E, and erecting given straight lines as in the problem DP perpendicular to FD; which gives the centre P we have just considered, it may be reWhen A is in one of the given lines, as BC, the solution takes a still more simple form, quired to describe a circle which shall I 2 E as before. |