8. To describe a circle which shall touch three given straight lines, not more than two of which are parallel.
Proceed as in Euclid IV. 4. If the given straight lines form a triangle, four circles can be described, namely, one as in Euclid, and three others each touching one side of the triangle and the other two sides produced. If two of the given straight lines are parallel, two circles can be described, namely, one on each side of the third given straight line.
9. To describe a circle which shall touch a given circle, and touch a given straight line at a given point.
Let A be the given point in the given straight line, and C be the centre of the given circle. Through C draw a straight line perpendicular to the given straight line,
and meeting the circumference of the circle at B and D, of which D is the more remote from the given straight line. Join AD, meeting the circumference of the circle at E. From A draw a straight line at right angles to the given straight line, meeting CE produced at F. Then F shall be the centre of the required circle, and FA its radius.
For the angle AEF is equal to the angle CED (I. 15); and the angle EAF is equal to the angle CDE (I. 29); therefore the angle AEF is equal to the angle EAF; therefore AF is equal to EF {I. 6).
