then all of the triangles are directly similar. Examples Triangles appended to a rectangle Direct similarity If we construct a rectangle with directly similar triangles on each side of the rectangle that are similar to , then is directly similar and the set of triangles is a similarity system. Indirect similarity However, if we acknowledge that the triangles can be degenerate and take points and to lie on each other and and to lie on each other, then the set of triangles is no longer a direct similarity system since the second triangle has area and the others do not. Rectangular parallelepiped Given a figure where three sets of lines are parallel, but not equivalent in length (formally known as a rectangular parallelepiped) with all points of order two being labelled as follows: Then we can take the above points, analyze them as triangles and we can show that they form a similarity system. Proof: In order for any given triangle, , to be directly similar to the following equation should be satisfied: where ℓ, m, k, a1, b1, and c1 are sides of triangles. If the same pattern is followed for the rest of the triangles, one will notice that the summation of the equations for the first four triangles and the summation of the equations for the last four triangles provides the same result. Therefore, by the definition of a similarity system of triangles, no matter the seven similar triangles selected, the eighth will satisfy the system, making them all directly similar. Gallery References Geometry Triangles Incidence geometry Configurations (geometry)