It is of course quite easy in practice to just place the straight edge in almost the right position by trial and error. And the process of going back and forth to get tangency on each circle by rotating around the contact point on the other does converge exponentially fast. But the Euclidean ideal is to get an exact placement of the ruler by steps which involve just placing it across exactly located points rather than sliding it around while trying to watch both ends at once. Here we show a fairly simple ruler and compass construction which gets the exact placement in a finite number of steps.
The use of a java applet (now displayed in html via webswing) makes it easy to create diagrams for several steps of this process and to do so for a whole range of possible cases of the sizes and positions of the circles.
This allows the user to quickly see that the claimed relationships are preserved and perhaps to gain some intuition as to why the construction works.