center of similitude | self-homologous point

If two similar figures lie in the plane but do not have parallel sides (i.e., they are similar but not homothetic), there exists a center of similitude, also called a self-homologous point, which occupies the same homologous position with respect to the two figures. The similitude center S of two triangles Δ A_1 A_2 A_3 and Δ B_1 B_2 B_3 can be constructed by extending each pair of corresponding sides of the triangles and locating their intersection, then drawing the circumcircle passing through two corresponding vertices of the triangles and the point of intersection of the pair of lines through corresponding sides that contain these points. Repeating for each of the three vertices gives three circles that intersect in a unique point, as illustrated above. This point is the similitude center S.

antihomologous points | circle-circle tangents | external similitude center | homologous points | homothetic center | internal similitude center | paralogic triangles | similar | similarity point | similitude circle