Circle construction

What is the essential difference, when constructing a circle, between the compass button and the circle construct button? Try the following simple exercise.

You will find that the correct construction can be made with either button. So how are the buttons different? Consider Euclid's second Proposition:

To place at a given point (as an extremity) a straight line equal to a given straight line.

Here are two exercises based on this proposition. The first gives you the compass tool to use, the second gives you the circle tool. Try them both.

Example 1

Example 2

If you have tried both exercises, you should find the first very easy indeed, and the second extremely hard. The solution to the second version requires looking at the problem as Euclid did. See Euclid's Proposition 2 for his solution. The first exercise above is easily solved because the compass tool not only draws circles but measures them too. This enables you to copy one circle to another position. Thus constructing a circle centred at A that has the same radius as BC is straightforward. These two capabilities are, of course, what all real compasses can do, and so these geometrical construction exercises will usually allow you to use the compass tool rather than the circle tool.

See also Circle-only constructions