11. To describe a circle which shall touch two given straight lines and a given circle.
Draw two straight lines parallel to the given straight lines, at a distance from them equal to the radius of the given circle, and on the sides of them remote from the centre of the given circle. Describe a circle touching the straight lines thus drawn, and passing through the centre of the given circle (7). A circle having the same centre as the circle thus described, and a radius equal to the excess of its radius over that of the given circle, will be the required circle.
Two solutions will be obtained, because there are two solutions of the problem in 7; the circles thus obtained touch the given circle externally.
We may obtain two circles which touch the given circle internally, by drawing the straight lines parallel to the given straight lines on the sides of them adjacent to the centre of the given circle.
12. To describe a circle which shall pass through a given point and touch a given straight line and a given circle.
We wall suppose the given point and the given straight line without the circle; other cases of the problem may be treated in a similar manner.
Let A be the given point, and B the centre of the given circle. From B draw a perpendicular to the given straight line, meeting it at C, and meeting the circumference of the given circle at D and E, so that D is between B and C. Join EA and determine a point F in EA produced if necessary, such that the rectangle EA, EF may be equal to the rectangle EC,ED; this can be done by describing a circle through A,C,D, which will meet EA at the required point (III. 36, Corollary). Describe a circle to pass through A and F and touch the given straight line (6); this shall be the required circle.
