PROPOSITION XIII. THEOREM. One circle cannot touch another at more points than one, whether it touch it internally or externally. First let the © ADE touch the © ABC internally at pt. A. Then there can be no other point of contact. B P Take O the centre of O ABC Then P, the centre of O ADE, lies in OA. III. 11. Take any pt. E in the Oce of the ✪ ADE, and join OE. Then from 0, a pt. within or without the C ADE, two lines OA, OE are drawn to the Oce, of which OA passes through the centre P, :: OA is greater than OE, and .. E is a point within the C ABC. III. 8, Cor. Post. Similarly it may be shewn that every pt. of the Oce of the O ADE, except A, lies within the ABC; .. A is the only point at which the Os meet. Next, let the Os ABC, ADE touch externally at the pt. A. Then there can be no other point of contact. Take O the centre of the O ABC. Then P, the centre of the ADE, lies in OA produced. III. 12. Take any pt. D in the Oce of the C ADE, and join OD. Then from O, a pt. without the O ADE, two lines OA, OD are drawn to the Oce, of which OA when produced passes through the centre P, .. A is the only point at which the Os meet. Q. E. D. DEF. VIII. The DISTANCE of a chord from the centre is measured by the length of the perpendicular drawn from the centre to the chord. Equal chords in a circle are equally distant from the centre; and conversely, those which are equally distant from the centre, are equal to one another. Let the chords AB, CD in the © ABDC be equal. and AB=CD, .. AP=CQ. Then. AP=CQ, and AO=CO, in the right-angled as AOP, COQ, .. OP=OQ; III. 3. I. E. Cor. p. 43. Def. 8. and.. AB and CD are equally distant from 0. Next, let AB and CD be equally distant from 0. Ex. In a circle, whose diameter is 10 inches, a chord is drawn, which is 8 inches long. If another chord be drawn, at a distance of 3 inches from the centre, shew whether it is equal or not to the former. The diameter is the greatest chord in a circle, and of all others that which is nearer to the centre is always greater than one more remote; and the greater is nearer to the centre than the less. E B Let AB be a diameter of the ABDC, whose centre is 0, and let CD be any other chord, not a diameter, in the O, nearer to the centre than the chord EF. Then must AB be greater than CD, and CD greater than EF. Draw OP, OQ 1 to CD and EF; and join OC, OD, OE. Then. AO=CO, and OB=OD, .. AB=sum of CO and OD, and.. AB is greater than CD. I. Def. 13. I. 20. Def. 8. Again, CD is nearer to the centre than EF, .: sum of sqq. on OP, PC=sum of sqq. on OQ, QE. I. 47. But sq. on OP is less than sq. on OQ; .. sq. on PC is greater than sq. on QE; and .. CD is greater than EF. Next, let CD be greater than EF. Then must CD be nearer to the centre than EF. Now the sum of sqq. on OP, PC=sum of sqq. on OQ, QE. But sq. on PC is greater than sq. on QE; .. sq. on OP is less than sq. on OQ; .. OP is less than OQ; and .. CD is nearer to the centre than EF. Q. E. D. Ex. 1. Draw a chord of given length in a given circle, which shall be bisected by a given chord. Ex. 2. If two isosceles triangles be of equal altitude, and the sides of one be equal to the sides of the other, shew that their bases must be equal. Ex. 3. Any two chords of a circle, which cut a diameter in the same point and at equal angles, are equal to one another. DEF. IX. A straight line is said to be a TANGENT to, or to touch, a circle, when it meets and, being produced, does not cut the circle. From this definition it follows that the tangent meets the circle in one point only, for if it met the circle in two points it would cut the circle, since the line joining two points in the circumference is, being produced, a secant. (III. 2.) DEF. X. If from any point in a circle a line be drawn at right angles to the tangent at that point, the line is called a NORMAL to the circle at that point. DEF. XI. A rectilinear figure is said to be described about a circle, when each side of the figure touches the circle. о And the circle is said to be inscribed in the figure. |