### Interactive Tools, Toys, and Games which support the learning and teaching of mathematics

This applet explores the geometric construction of a line tangent to both of two given circles by a finite sequence of steps all of which can be done with just the basic Euclidean tools of ruler and compass. 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. There is in fact a fairly simple ruler and compass construction which gets the exact placement in a finite number of steps. The use of a java applet 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.

This applet explores the geometric theorem that for any three equilateral triangles which share a common vertex, the midpoints of the lines joining non-shared vertices of pairs of adjacent triangles themselves always form an equilateral triangle. This allows the user to quickly see that the claimed relationships are preserved and perhaps to gain some intuition as to why the theorem is true.

If a picture is hanging above your eye level, then if you get up close to the wall you only see it take up a very small angle from top to bottom in your field of vision. And it also appears very small from far away. So it seems natural to ask what distance gives the biggest such angle. One way to answer that question is to use calculus, and this kind of problem is often used as an example or exercise in calculus classes. The result is that the viewer’s distance from the wall should be the geometric mean (ie square root of the product) of the heights above eye level of the top and bottom of the picture. But this result can also be proved by using nothing more than Euclidean Geometry.

Check out this applet for a walk through the proof.