5. To describe a circle which shall pass through three given points not in the same straight line.
This is solved in Euclid IV. 5.
6. To describe a circle which shall pass through two given points on the same side of a given straight line, and touch that straight line.
Let A and B be the given points; join AB and produce it to meet the given straight line at C. Make a square equal to the rectangle CA, CB (II. 14), and on the
given straight line take CE equal to a side of this square.
Describe a circle through A, B, E (5); this will be the circle required (III. 37).
Since E can be taken on either side of C, there are two solutions.
The construction fails if AB is parallel to the given straight line. In this case bisect AB at D, and draw DC at right angles to AB, meeting the given straight line at C.
Then describe a circle through A, B, C.
7. To describe a circle which shall pass through a given point and touch two given straight lines.
Let A be the given point; produce the given straight lines to meet at B, and join AB. Through B draw a straight line, bisecting that angle included by the given straight lines within which A lies; and in this bisecting straight line take any point C. From C draw a perpendicular on one of the given straight lines, meeting it at D; with centre C, and radius CD, describe a circle, meeting AB, produced if necessary, at E. Join CE; and through A draw a straight line parallel to CE, meeting BC, produced if
