20. A and B are two given points on the same side of a given straight line, and AB produced meets the given straight line at C; of all points in the given straight line on each side of C, it is required to determine that at which AB subtends the greatest angle.
Describe a circle to pass through A and B, and to touch the given straight line on that side of C which is to be considered (6). Let D be the point of contact: D shall be the required point.
For, take any other point E in the given straight line, on the same side of C as D is; draw EA, EB; then one at least of these straight lines will cut the circumference ADB.
Suppose that BE cuts the circumference at F; join AF. Then the angle AFB is equal to the angle ADB (III. 21); and the angle AFB is greater than the angle AEB (I. 16); therefore the angle ADB is greater than the angle AEB.
21. A and B are two given points within a circle; and AB is drawn and produced both ways so as to divide the whole circumference into two arcs; it is required to determine the point in each of these arcs at which AB subtends the greatest angle.
