Prove: Exercise 3.22
Proof: Exercise 3.22 a
Suppose we are given line segment AB. We then construct two circles X and Y -- X centered at A with radius AB and Y centered at B with radius BA. Let C and D be the intersections of X and Y. Let E be the point for which AB and CD intersect. We wish to prove that ABC ABD.
Note that by the definition of a circle, we know that AC AB. Similarly, we know that BA BC. Also, since AB BA, it follows by CN1 that AC BC. Therefore, we have that AC BC AB. By the definition of equilateral triangle, we have that ABC is an equilateral triangle.
Analogously, by the definition of circle, we know that AB AD and BA BD. Also, since AB BA, it follows by CN1 that AD BD. Therefore, AD BD AB. Once more, by the definition of equilateral triangle, we have that ABD is an equilateral triangle. Since both ABC and ABD contain side AB, by Proposition 8, we have that ABC ABD.