therefore AG is less than AC; therefore G is inside the circle whose centre is A. Hence the circles intersect. I. 13, Cor. III. 1, Cor. III. Def. 14. Again, the line AB bisects the line CD at right angles, by construction; and AB is less than the sum of AC and BC, and greater than the difference of AC and BC. I. 13. I. 13, Cor. COR. If two circles meet in one point only, that point lies on the line joining the centres. For it has been shown that if the circles have a common point not on the line joining their centres, then they must meet in more than one point hence, by contraposition, if the circles meet in one point only, that point lies on the line joining their centres. THEOR. 25. If two circles meet in a point which is on the line joining their centres they do not meet in any other point, but touch either externally or internally; and the distance between their centres is in the former case equal to the sum, and in the latter to the difference, of the radii. Let the circumferences of two circles whose centres are A and B meet at the point C on the straight line through A and B : Fig. 1. B they shall not meet in any other point. Fig. 2. B Take any other point D on the circumference of the circle whose centre is B; join AD and BD. Then in fig. I the sum of AD and BD is greater than AB, and BD is equal to BC; I. 13. Ax. g. therefore AD is greater than AC; therefore D is outside the circle whose centre is A; III. 1, Cor. therefore the circle whose centre is B is wholly outside the circle whose centre is A; therefore the circles touch externally. III. Def. 14. I. 13. In fig. 2 AD is less than the sum of AB and BD, and BD is equal to BC; therefore AD is less than AC; therefore D is within the circle whose centre is A; therefore the circle whose centre is B is wholly within the circle whose centre is A ; therefore the circles touch internally. III. Def. 14. Also in fig. I AB is equal to the sum, and in fig. 2 to the difference, of the radii AC and BC. Q.E.D. COR. I. Hence by contraposition, if two circles intersect, neither point is on the line joining their centres. COR. 2. The Cor. to Theor. 24 may also be stated thus: If two circles touch one another (whether internally or externally) the line joining their centres passes through the point of contact. COR. 3. If two circles touch one another, they have a common tangent at the point of contact. For the straight line drawn from the point of contact perpendicular to the radius of one circle will be perpendicular to the radius of the other, and will therefore be a tangent to both circles. III. 20, Cor. 2. *Ex. 98. If two circles do not meet and lie each wholly outside the other, the distance between their centres is greater than the sum of their radii; and if one lies wholly inside the other, that distance is less than the difference of their radii. *Ex. 99. According as the distance between the centres of two circles is (1) greater than the sum of the radii, (2) equal to that sum, (3) less than the sum but greater than the difference of the radii, (4) equal to the difference of the radii, (5) less than the difference; so do the circles (1) lie each wholly outside the other, (2) touch each other externally, (3) intersect, (4) touch each other internally, (5) lie one inside the other. Shew that these statements follow from the preceding statements by the Rule of Conversion, and give also a direct geometrical proof of each. PROB. 5. To draw a common tangent to two given circles. Let A and B be the centres of the given circles of which the former is not the less: it is required to draw a common tangent to the circles, B B With centre A, and radius equal to the sum or difference of Post. 3. III. 22, Cor. 2. the radii of the given circles, describe a circle; from B draw BC to touch this circle at C ; join AC, and let AC, or AC produced through C, meet the circumference of the circle whose centre is A at D ; through B draw BE parallel to CD, and on the same side of BC as CD, to meet the circumference of the circle whose centre is B at E. I. Prob. 6. Join DE: then shall DE be a common tangent to the given circles. Because BC is a tangent to the circle through C, therefore the angle ACB is a right angle. III. 20. Again, AC is equal to the sum or difference of AD and BE; therefore BE is equal to CD, and BE is parallel to CD; therefore BCDE is a parallelogram; I. 31. and BCD one of its angles is a right angle; therefore the angles CDE and BED are also right angles; therefore DE touches both the given circles. I. 28, Cor. III. 20. Q.E.F. If the given circles are wholly outside each other, AB is greater than the sum of their radii, and B lies outside each of the circles used in the construction, and four common tangents can be drawn. If the circles touch each other externally, AB is equal to the sum of their radii, and B lies on the circumference of one of the circles used in the construction and outside the other, and three common tangents can be drawn ; if the circles intersect, AB is less than the sum and greater than the difference of their radii, and B lies within one and outside the other of the circles used in the construction, and two common tangents can be drawn. If the circles touch internally, AB is equal to the difference of their radii, and B lies within one, and on the circumference of |