Triangle ABC is the original triangle. Drag the vertices of ABC to change the shape of ABC and its copies.
Given the triangle ABC, start with point D and the ray from D. Then use the compass (which draws the circles) to draw a circle with center D of radius AB; the intersection of the circle with the ray is E. Then draw a circle with center D and radius AC and a circle with center E and radius BC. The two intersection points are F and G. Then triangle ABC has sides of the same length as triangle DEF since the side lengths of DEF are radii of the circles. So ABC is congruent to DEF.
Likewise ABC is also congruent to triangle DEG. One of the triangles DEF and DEG has the same orientation as ABC (namely, it is related to ABC by a rotation or a translation) and the other has the opposite orientation.
Page by James King, 10/2002.