12. A series of circles touch a given straight line at a given point: shew that the tangents to them at the points where they cut a given parallel straight line all touch a fixed circle, whose centre is the given point. 13. If two circles touch one another internally, and any third circle be described touching one internally and the other externally ; then the sum of the distances of the centre of this third circle from the centres of the two given circles is constant. V 14. Find the locus of points such that the pairs of tangents drawn from them to a given circle contain a constant angle. V 15. Find a point such that the tangents drawn from it to two given circles may be equal to two given straight lines. When is this impossible? 16. If three circles touch one another two and two; prove that the tangents drawn to them at the three points of contact are concurrent and equal. THE COMMON TANGENTS TO Two CIRCLES. 17. To draw a common tangent to two circles. First. When the given circles are external to one another, or when they intersect. Let A be the centre of the greater circle, and B the centre of the less. From A, with radius equal to the diffce of the radii of the given circles, describe a circle: and from B draw BC to touch the last drawn circle. Join AC, and produce it to meet the greater of the given circles at D. Through B draw the radius BE par? to AD, and in the same direction. Join DE. Constr. Constr. :: DE is equal and parl to CB. 1. 33. But since BC is a tangent to the circle at C, .. the L ACB is a rt. angle ; III. 18. hence each of the angles at D and E is a rt. angle: 1. 29. .: DE is a tangent to both circles. Q.E.F. a It follows from hypothesis that the point B is outside the circle used in the construction : :: two tangents such as BC may always be drawn to it from B; hence two common tangents may always be drawn to the given circles by the above method. These are called the direct common tangents. Secondly. When the given circles are external to one another and do not intersect, two more common tangents may be drawn. For, from centre A, with a radius equal to the sum of the radii of the given circles, describe a circle. From B draw a tangent to this circle ; and proceed as before, but draw BE in the direction opposite to AD. It follows from hypothesis that B is external to the circle used in the construction ; :: two tangents may be drawn to it from B. Hence two more common tangents inay be drawn to the given circles : these will be found to pass between the given circles, and are called the transverse common tangents. Thus, in general, four common tangents may be drawn to two given circles. The student should investigate for himself the number of common tangents which may be drawn in the following special cases, noting in each case where the general construction fails, or is modified : (i) When the given circles intersect : (ii) When the given circles have external contact: (iii) When the given circles have internal contact : (iv) When one of the given circles is wholly within the other. 8 18. Draw the direct common tangents to two equal circles. 19. If the two direct, or the two transverse, common tangents are drawn to two circles, the parts of the tangents intercepted between the points of contact are equal. 20. If four comnion tangents are drawn to two circles external to one another; shew that the two direct, and also the two transverse, tangents intersect on the straight line which joins the centres of the circles. 21. Two given circles have external contact at A, and a direct common tangent is drawn to touch them at P and Q: shew that PQ subtends a right angle at the point A. 22. Two circles have external contact at A, and a direct common tangent is drawn to touch them at P and Q: shew that a circle described on PQ as diameter is touched at A by the straight line which joins the centres of the circles. a 23. Two circles whose centres are Cand C' have external contact at A, and a direct common tangent is drawn to touch them at P and Q : shew that the bisectors of the angles PCA, QC'A ineet at right angles in PQ. And if R is the point of intersection of the bisectors, shew that RA is also a conimon tangent to the circles. 24. Two circles have external contact at A, and a direct common tangent is drawn to touch them at P and Q: shew that the square on PQ is equal to the rectangle contained by the diameters of the circles. 25. Draw a tangent to a given circle, so that the part of it intercepted by another given circle may be equal to a given straight line. When is this impossible? 26. Draw a secant to two given circles, so that the parts of it intercepted by the circumferences may be equal to two given straight lines. PROBLEMS ON TANGENCY. Obs. The following exercises are solved by the Method of Intersection of Loci, explained on page 125. The student should begin by making himself familiar with the following loci. (i) The locus of the centres of circles which pass through two given points. (ii) The locus of the centres of circles which touch a given straight line at a given point. (iii) The locus of the centres of circles which touch a given circle at a given point. (iv) The locus of the centres of circles which touch a given straight line, and have a given radius. (v) The locus of the centres of circles which touch a given circle, and have a given radius. (vi) The locus of the centres of circles which touch two given straight lines. In each exercise the student should investigate the limits and relations among the data, in order that the problem may be possible. 27. Describe a circle to touch three given straight lines. 28. Describe a circle to pass through a given point, and touch a given straight line at a given point. 29. Describe a circle to pass through a given point, and touch a given circle at a given point. a a 30. Describe a circle of given radius to pass through a given point, and touch a given straight line. 31. Describe a circle of given radius to touch two given circles. 32. Describe a circle of given radius to touch two given straight lines. 33. Describe a circle of given radius to touch a given circle and a given straight line. 34. Describe two circles of given radii to touch one another and a given straight line, on the same side of it. 35. If a circle touches a given circle and a given straight line, shew that the points of contact and an extremity of the diameter of the given circle at right angles to the given line are collinear. 36. To describe a circle to touch a given circle, and also to touch a given straight line at a given point. Let DEB be the given circle, PQ the given straight line, and A the given point in it. It is required to describe a circle to touch the O DEB, and also to touch PQ at A. At A draw AF perp. to PQ:1.11. then the centre of the required circle must lie in AF. 111. 19. Find C, the centre of the O DEB, B JII. 1. and draw a diameter BD perp. to PQ: P A Join CE, and produce it to cut AF at F. [Supply the proof : and shew that a second solution is obtained by joining AB, and producing it to meet the Oce. Also distinguish between the nature of the contact of the circles, when PQ cuts, touches, or is without the given circle.] 37. Describe a circle to touch a given straight line, and to touch a given circle at a given point. 38. Describe a circle to touch a given circle, have its centre in a given straight line, and pass through a given point in that straight line. [For other problems of the same class see page 253.] ORTHOGONAL CIRCLES. DEFINITION. Circles which intersect at a point, so that the two tangents at that point are at right angles to one another, are said to be orthogonal, or to cut one another orthogonally. L 39. In two intersecting circles the angle between the tangents at one point of intersection is equal to the angle between the tangents at the other. 40. If two circles cut one another orthogonally, the tangent to each circle at a point of intersection will pass through the centre of the other circle. 41. If two circles cut one another orthogonally, the square on the distance between their centres is equal to the sum of the squares on their radii. 42. Find the locus of the centres of all circles which cut a given circle orthogonally at a given point. 43. Describe a circle to pass through a given point and cut a given circle orthogonally at a given point. III. ON ANGLES IN SEGMENTS, AND ANGLES AT THE [See Propositions 20, 21, 22 ; 26, 27, 28, 29; 31, 32, 33, 34.] 1. If two chords intersect within a circle, they form an angle equal to that at the centre, subtended by half the sum of the arcs they cut off. Let AB and CD be two chords, intersecting at E within the given O ADBC. Then shall the L AEC be equal to the angle at the centre, subtended by half the sum of the B arcs AC, BD. Join AD. Then the ext. L AEC=the sum of the int. A opp. 28 EDA, EAD; that is, the sum of the L® CDA, BAD. But the 4* CDA, BAD are the angles at the Oce subtended by the arcs AC, BD; :: their sum=half the sum of the angles at the centre subtended by the same arcs ; or, the L AEC=the angle at the centre subtended by half the sum of the arcs AC, BD. Q. E.D. |