7. If possible, let a st. line cut the o in the three points A, E, B, whereof E is between A and B. Then E must fall within the O. But it was assumed to be on the circumference. Hence the st. line AB cannot cut the o in more than two points. 8. Draw a chord perp. to the st. line joining the given pt. to the centre. This chord will be bisected at the given pt. 9. Let o be the common centre, ABCD the st. line cutting the inner o in B, c, and the outer o in A, D. Draw OX perp. to ABCD. Then BX=CX and AX = DX. ... difference AB = difference CD. 10. The st. line, through the centre, perp. to one of the par! chords, is perp. to the other [1. 29]. And this st. line bisects both chords. Hence, the st. line joining the middle pts. of two par!. chords passes through the centre. 11. The st. line, through the centre, perp. to one of the part. chords, is perp. to all of them [1. 29]. And this st. line bisects all the chords. Hence it is the required locus. 12. Let the two os intersect in A, B. Let CAD, EBF be parł. st. lines cutting the one circle in C, E and the other in D, F. Then the st. line through the centre of ABEC perp. to the chords AC, BE bisects these chords in P and Q, say. Similarly the st. line through the centre of ABFD perp. to the chords AD, BF bisects these chords in X and Y, say. But PQ is par! to XY (1. 28]. :. PX = QY (1. 34]. But CA is double of PA, and AD is double of AX. .. CD is double of Px. Similarly EF is double of QY, .. CD = EF. 13. Bisect PQ in R, and XY in Z. Let PY, QX intersect in 0. Join OR, OZ. Then APXY = AQXY: and AXOZ = A YOZ. .. APXO=A QYO. Also A POR = A QOR. .. AS POR, PXO, XOZ together =A8 QOR, QYO, YOZ: i.e. the lines OR, oz bisect the trapezium PXYQ. But the st. line Rz bisects the trapezium PXYQ [Ex. 8, p. 109, 1. 38]. .. the st. line RZ coincides with the st. lines OR, OZ: that is, o lies on RZ. Similarly if PX, QY intersect in o', o' lies on RZ. Page 158. 1. The diagonals of a parm, bisect one another... their pt. of intersection is the centre. 2. Let the parm. ABCD be inscribed in a 0. Then the diagonals AC, BD intersect in 0, the centre of the o [Ex. 1]. Because AO=BO, .. LOAB LOBA. And because AO= DO, .LOAD = LODA; ... DAB is a rt. 2 (1. 32]. 3. Let C, D be the centres of two os intersecting in A. Draw AX perp. to CD, and produce it to B, so that BX = AX. Then CA=CB and DA= DB (1. 4]. .. B is a pt. on both circles. Page 168. 1. Let A be the centre of the larger, B of the smaller O. Produce AB to C, making BC = the radius of the smaller O. Then AC is the radius of the larger. .. the Os meet at C. Let D be any other pt. on the smaller 0. Then BD=BC. .. AB, BD together= AC. But AB, BD together > AD. .. AC > AD. ..D cannot be on the circle with centre A and radius AC. 2. Let c be the pt. of contact. Then ABC is a st. line [111. 11]. Because AC = AP, therefore 2 ACP = L APC. And because BC =BQ, .. LBCQ = BQC... LAPC = L BQC. .. AP, BQ are parallel. Page 169. 1. The required locus is the st. line joining the centre of the given o to the given pt. [111. 11, 12]. 2. Let o be the centre of the given 0, OP its radius. On OP take PC equal to the given radius of the circles which are to touch the given O. Then the O with centre C and radius CP will touch the given o at P. And oC = the sum or the difference of OP and CP. Hence the required locus is a circle with centre o and radius equal to the sum or the difference of the radius of the given circle and the given radius of the touching circles. 3. Let A, B be the centres of the two circles. From AB cut off AC = radius of O with centre A. Then BC = radius of 0 with centre B... the Os meet at C. Let D be any other pt. on O with centre B. Then AD, DB together > AB. But BD = BC. .. AD > AC. .:. D cannot be on the circle with centre A and radius AC. 4. Let c be the pt. of contact. Then ACB is a st. line (III. 12]. Because AC = AP, :: ACP = L APC. And because BC = BQ, .. LBCQ= L BQC. .. L APC=LBQC. .. AP, BQ are parallel. Page 171. 1. Let A, B be the two given pts. Bisect AB in C: draw CX perp. to AB. Then, if cx coincides with the given st. line, with any pt. X on CX as centre, the circle described with centre C and radius CA will pass through B. But if cx is par!. to the given st. line no circle can be described as required. Finally, if cx cuts the given st. line in X, the circle described with centre X and radius CA or CB is the circle required. 2. Let A be the given pt., XY the given st. line. Draw AM perp. to XY, and produce it to B, so that BM = AM. All the circles will pass through B. (See Ex. 1, p. 215.] 3. Let A be the centre of the given O, P the given pt. Take PC equal to the given radius, either on PA (produced if necessary) or on AP produced. The circles described with centre c and radius CP will touch the given circle at P: but, if the given radius is equal to the radius of the given circle, one of the two circles so described will coincide with the given circle. 4. Let A be the centre of the given 0, B the given pt. Let AB cut the given o in C and D. The o described with centre B and radius BC or BD will touch the given O. Hence there are two solutions except when B is on the oce of the given O. 5. Let A be the centre of the given 0, B the given pt. on it, c the given pt. through which the required o is to pass. Let the perp. bisector of BC cut AB in 0. The o described with centre o and radius OB is the o required : but, if c is on the circumference of the given 0, 0 will coincide with a [111. 1], and the o described will coincide with the given O. The solution is also impossible, if CB is perp. to AB. For then the perp. bisector of BC will not cut AB. 6. Let A and B be the centres of the two given OS X and Y. Describe a circle with centre A and radius equal to the sum or the difference of the radius of the required circle and the radius of X. Describe a circle with centre B and radius equal to the sum or the difference of the radius of the required circle and the radius of y. Then the circle described with either of the pts. of intersection of these two circles as centre and with the required radius will be the circle required. There are thus in general 8 possible solutions. 7. Let A, B be the middle pts. of the given chords DAF, EBG, so that AD = 3 in. and Be= 4 in. Then, supposing the chords to be on the same side of the centre, let AB be produced to C the centre. Then CD2=CE”; that is, CAR + 9 = CB? + 16. .. CAP – CB2= 7. But CA - CB=1. :. CA + CB, that is, AB + 2CB=7. .. CB = 3. ::. CE= CBP + BE? = 32 + 42 = 25. 1. CE (the radius) = 5. 8. Let A, B be the centres of the two circles, touching externally at C. Then ACB is a st. line [111. 12]. Draw the parallel diameters DAE, FBG. Then, because AD= AC, .. LADC = L ACD, .. ext. _ EAC = twice - ACD. Similarly, L CBF = twice - BCG. But L EAC = L CBF, because AE, BF are par! . LACD _ BCG. :: DC, CG are in a st. line; that is, the st. line joining GD passes through c. 9. Let A, B be the centres of the two Os; and D, E the pts. where AB cuts the circles. Let PQ be any other st. line cutting the circles in P and Q. Let QA cut the 'A' circle in R. Then QP > QR (111. 8). Also AQ> AE, of which AR = AD; .. QR> DE. .. à fortiori, QP > DE. :. DE is less than any st. line PQ cutting the Os in P and Q. Similarly, if AB produced cut the circles in FG, FG is greater than any of the st. lines PQ cutting the OS in P and Q. 10. Let BC be a chord : A the centre, and ADE the radius bisecting BC at rt. _ $ in D. Let G be any pt. in BD, and GF perp. to BD, cutting the arc Bec in F. Draw AH perp. to FG. Then AE = AF > HF. And AD = HG; .. the whole or remainder DE > whole or remainder GF. In DG take a pt. L, and draw MLK perp. to BC to meet the arc at K. Then AKP = AF?, that is, AMP + MK2= AH2 + HF? But AH?> AM?. .. HF < MK”, that is, GF <LK. 11. Let A be any pt. on the circumference : ACB the diameter, AD any other chord. Then CB=CD; .. AB = AC + CD > AD. And if AF is nearer to AB than AD, the two sides AC, CD are equal to the two sides AC, CF, but the angle ACF > angle ACD, .. base AF > AD [1. 24]. Make - ACE = L ACF. Then AE = AF (1. 4). But AF > AD. Hence two and only two chords from A can be drawn equal to one another. Page 173. 1. Since equal chords are equidistant from the centre, the locus is a circle, whose centre is the centre of the given circle and radius is equal to the distance of any of the chords from the centre. 2. Let chords AB, CD cut in E. Let F be the centre. Draw FG, FH perp. to AB, CD. Then in the right-angled AS FEG, FEH, the hypotenuse EF is common and the angles GEF, HEF are equal. .. GF=HF [1. 26], -. the chords AB, CD are equidistant from the centre, and are therefore equal (111. 14]. 3. Take fig. of preceding Ex. Then EG = EH. But BG, HD, the halves of AB, CD [111. 3], are equal. .. the whole or remainder BE = whole or remainder CE. 4. With centre, A, on the o ce of the given 0, describe a circle having radius equal to the required length, cutting the given circle again in B. From the centre of the given circle, draw a st. line perp. to the required direction and equal to the distance of the centre from AB. The chord drawn through the extremity of this st. line, par! to the given direction, will be the chord required. 5. Let o be the centre; and AX, BY, OZ the perpø. on PQ. Then Oz = { the sum or difference of AX and By, according as A and B are on the same or opposite sides of PQ [Ex. 18, p. 98]. .. the sum or difference of AX and BY = twice OZ = a constant. Page 175. 1. Let A be the given pt., and B the centre. Draw the chord CAD perp. to AB, and any other chord EAF through A. Draw BG perp. to EF. Then the hypotenuse BA > BG. .. chord CD < chord EF; that is, CD is the least chord through A. H. K. E. 4 |