## Isometries product

Here we are interested in a special product of isometries, defined as follows. Given a qudrangle t = (ABCD) and a point J on its side AB, defining the translation s = [2*BJ]. We are interested in the product of isometries f = c*d*a*s*b. Here a, b, c, d indicate the clockwise rotations about the vertices A, B, C, D correspondingly. The composition of isometries is applied from right to left: ( f(x) = c(...(b(x)...). Such a product is always a translation by the vector (2*WX), parallel to the side BC.
The proof is essentially the same with that given in RotationsOnQuadrangleVertices.html .

The translation s is handled like the corresponding s appearing in the file Isometries_product.html . The triangle (OAQ) is similar to triangle (ABN) formed by the bisectors at angles A and B. (OAQ) has O glidding on line OB (external bisector of angle B) and Q glidding on the bisector of the angle formed by the prolongations of the sides AD and BC. Therefore (SPTQ) is inscribable in a circle.

A nice application of this result is discussed in the file Rotation_product_odd.html .