Constructing a Rhombus

Pick two points A and B. Draw a circle centered on A, through B. Pick another point C on this circle. Draw a circle \beta centered on B through A, and circle \gamma centered on C through A. These circles intersect at two points, one of which is A, the other is new point D. AB and AC are both radii of the same circle \alpha, therefore congruent. Segment AB is a radius of both circles \alpha and \beta, so they are congruent also. Likewise circle \alpha and \gamma. As a result, circles \beta and \gamma are also congruent. Point D is on both circles \beta and \gamma, and since \beta and \gamma are congrent, segments BD and CD are the same length. All the circles have the same radius, and all the segments are radii, so all of the segments are congruent. The polygon ABDC is a rhombus.