And consequently the line joining ED must be parallel to C M. Similarly, BD: MDR: R3, BF:CFR: R1; therefore B D:MD=BF:CF, And consequently the line joining FD must be parallel to C M. Hence F, E, and D must lie in the same straight line. 25. Prove that each of the points of intersection of interior common tangents lies in the same straight line with two points of intersection of exterior common tangents. 3 Let A, B, and C be the centres of the circles, and R1, R2, R, the radii. Let interior common tangents to A and C intersect in E, and to B and C in F; and let the exterior common tangents to A and B intersect in D. With A and B as centres and R1 + R, and R2+R, as radii, describe circles, and draw their common tangent, H L, meeting the line A B in M, and then draw tangents from C. Join C M, as in the last example. AE: ECR1: R1, AD: DM=R1: R1; therefore AE:EC=AD: DM. Hence the line joining ED will be parallel to CM. ヨ B Similarly, the line joining F D will be parallel to C M. Consequently E, F, and D lie in the same straight line. 26. Given the radii of the inscribed and circumscribed circles of an isosceles triangle to construct it. Let A B C be an isosceles triangle inscribed in a circle, B and C being the equal angles. Draw the diameter A F, and let D be centre of the inscribed circle; also draw DE perpen dicular to A C, and join D C, Moreover, since the = FAC; also the angle BCD ACD. Hence / FCD =(DAC+ AC D) = F DC, consequently FD= FC. triangles ADE, AFC are similar, A D:AF= DE: FC or F D, therefore AD· FD=AF·DE, AF and DE being respectively the diameter and radius of the circumscribed and inscribed circles. At D draw a perpendicular to A F to meet the circle in H, and from H draw a line parallel to AF to meet the tangent at F in G. Then FGDH2 AD FD=AF D E. Construction.-Let r, R represent the respective given radii. Make A F = 2 R, and F G perpendicular to A F and equal to 2 R · r. Draw from G a line parallel to A F to meet the circle of which AF is a diameter in H, and draw HD perpendicular to AF. From F draw to the circle F C, FB, each equal to FD, and draw A B, BC, CA, which lines form the triangle required. NOTE. When the line drawn from G parallel to AF cuts the circle, two triangles may be found; but when this line touches the circle, there exists but one. |