4. To draw a straight line which shall touch two given circles.
Let A be the centre of the greater circle, and B the centre of the less circle. With centre A, and radius equal to the difference of the radii of the given circles, describe a circle; from B draw a straight line touching the circle
so described at C. Join AC and produce it to meet the circumference at D. Draw the radius BE parallel to AD and on the same side of AB; and join DE. Then DE shall touch both circles.
See I. 33, I. 29, and III. 16 Corollary.
Since two straight lines can be drawn from B to touch the described circle, two solutions can be obtained; and the two straight lines which are thus drawn to touch the two given circles can be shewn to meet AB, produced through B, at the same point. The construction is applicable when each of the given circles is without the other, and also when they intersect.
When each of the given circles is without the other we can obtain two other solutions. For, describe a circle with A as a centre and radius equal to the sum of the radii of the given circles; and continue as before, except that BE and AD will now be on opposite sides of AB. The two straight lines which are thus drawn to touch the two given circles can be shewn to intersect AB at the same point.
