13. To describe a circle which shall touch a given straight line and two given circles.
Let A be the centre of the larger circle and B the centre of the smaller circle. Draw a straight line parallel to the given straight line, at a distance from it equal to the radius of the smaller circle, and on the side of it remote from A. Describe a circle with A as centre, and radius equal to the difference of the radii of the given circles. Describe a circle which shall pass through B, touch externally the circle just described, and also touch the straight line which has been drawn parallel to the given straight line (12). Then a circle having the same centre as the second described circle, and a radius equal to the excess of its radius over the radius of the smaller given circle, will be the required circle.
Two solutions will be obtained, because there are two solutions of the problem in 12; the circles thus described touch the given circles externally.
We may obtain in a similar manner circles which touch the given circles internally, and also circles which touch one of the given circles internally and the other externally.
14. Let A be the centre of a circle, and B the centre of a larger circle; let a straight line be drawn touching the former circle at C and the latter circle at D, and meeting AB produced through A at T. From T draw any straight line meeting the smaller circle at K and L, and the larger circle at M and N; so that the five letters T, K, L, M, N are in this order. Then the straight lines AK, KC, CL, LA shall be respectively parallel to the straight lines BM, MD, DN, NB; and the rectangle TK, TN shall he equal to the rectangle TL, TM, and equal to the rectangle TC, TD.
Join AC, BD. Then the triangles TAC and TBD are
