Bridging the Gap...Or not
Below is Euclid's proof for Proposition 1: constructing an equilateral triangle. Follow the steps and compare your results with his. 1. Construct a circle with radius AB centered at A. 2. Construct another circle with radius AB centered at B. 3. Plot a point of intersection of the circles. Label this point C. 4. Construct equilateral triangle ABC. 5. Verify that each of these objects are rational objects. 6. By now, you should have noticed that the equilateral triangle you constructed does not exist. Why not?