Euclid's Proposition 1: On a given segment, an equilateral triangle can be constructed.

On a given segment, an equilateral triangle can be constructed.

1. Let one endpoint of the segment be A. Let the other endpoint be B.
2. By Euclid's postulate 3, a circle may be described with any point as center and any distance as radius. Draw a circle with A as the center and a radius the same as the length of segment AB.
3. Now draw a circle with B as the center and a radius the same length as segment BA.
4. Since all points on a circle are the same distance from the center, and the radius is the same as the length of segment AB, the two intersection points of the circle are not only the same distance from A and B, but that distance is the distance between A and B.
5. Place a point C at one of the intersections of the circle.
6. Connect points A and B to C. The resulting triangle is an equilateral triangle.