II. ON THE TANGENT AND THE CONTACT OF CIRCLES. [See Propositions 11, 12, 16, 17, 18, 19.] 1. All equal chords placed in a given circle touch a fixed con centric circle. 2. If from an external point two tangents are drawn to a circle, the angle contained by them is double the angle contained by the chord of contact and the diameter drawn through one of the points of contact. 3. Two circles touch one another externally, and through the point of contact a straight line is drawn terminated by the circumferences: shew that the tangents at its extremities are parallel. 4. Two circles intersect, and through one point of section any straight line is drawn terminated by the circumferences: shew that the angle between the tangents at its extremities is equal to the angle between the tangents at the point of section. 5. Shew that two parallel tangents to a circle intercept on any third tangent a segment which subtends a right angle at the centre. 6. Two tangents are drawn to a given circle from a fixed external point A, and any third tangent cuts them produced at P and Q: shew that PQ subtends a constant angle at the centre of the circle. 7. In any quadrilateral circumscribed about a circle, the sum of one pair of opposite sides is equal to the sum of the other pair. 8. If the sum of one pair of opposite sides of a quadrilateral is equal to the sum of the other pair, shew that a circle may be inscribed in the figure. [Bisect two adjacent angles of the figure, and so describe a circle to touch three of its sides. Then prove indirectly by means of the last exercise that this circle must also touch the fourth side.] 9. Two circles touch one another internally, the centre of the outer being within the inner circle: shew that of all chords of the outer circle which touch the inner, the greatest is that which is perpendicular to the straight line joining the centres. 10. In any triangle, if a circle is described from the middle point of one side as centre and with a radius equal to half the sum of the other two sides, it will touch the circles described on these sides as diameters. 11. Through a given point, draw a straight line to cut a circle, so that the part intercepted by the circumference may be equal to a given straight line. In order that the problem may be possible, between what limits must the given line lie, when the given point is (i) without the circle, (ii) within it? 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. 14. Find the locus of points such that the pairs of tangents drawn from them to a given circle contain a constant angle. 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 В draw the radius BE par1 to AD, and in the same direction. Join DE. Then DE shall be a common tangent to the two given circles. For since AC the diffee between AD and BE, But since BC is a tangent to the circle at C, .. the L ACB is a rt. angle; hence each of the angles at D and E is a rt. angle: Constr. Constr. I. 33. III. 18. I. 29. Q.E.F. 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 may 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. 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 common 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. 23. Two circles whose centres are C and 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, QCA meet at right angles in PQ. And if R is the point of intersection of the bisectors, shew that RA is also a common 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. 30. Describe a circle of given radius to pass through a given point, and touch a given straight line. 31. 32. lines. Describe a circle of given radius to touch two given circles. Describe a circle of given radius to touch two given straight 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 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. III. 19. Then F shall be the centre, and FA the radius of the required circle. [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.] |