In a similar manner another solution may be obtained by joining AB. If the given straight line falls without the given circle, the circle obtained by the first solution touches the given circle externally, and the circle obtained by. the second solution touches the given circle internally. If the given straight line cuts the given circle, both the circles obtained touch the given circle externally.
10. To describe a circle which shall pass through two given points and touch a given circle.
Let A and B be the given points. Take any point C on the circumference of the given circle, and describe a circle through A, B, C. If this described circle touches the given circle, it is the required circle. But if not, let D
be the other point of intersection of the two circles. Let AB and CD be produced to meet at E; from E draw a straight line touching the given circle at F. Then a circle described through A, B, F shall be the required circle.
See III. 35 and III. 37.
There are two solutions, because two straight lines can be drawn from E to touch the given circle.
If the straight line which bisects AB at right angles passes through the centre of the given circle, the construction fails, for AB and CD are parallel. In this case F must be determined by drawing a straight line parallel to AB so as to touch the given circle.
