Isometries_product

Products of isometries

Here we are interested in a special product of isometries, defined as follows. Given a triangle t = (ABC) and a point E on its side BC, defining the translation s = [2*CE]. We are interested in the product of isometries f = a*b*s*c. Here a, b, c indicate the clockwise rotations about the vertices A, B, C, correspondingly. The composition of isometries is applied from right to left: ( f(x) = a(b(s(c(x))))). Such a product is always a point-symmetry on some point K of the side AC.
This is proved by the previous figure and the remarks:
- c is the product of reflexions on CF and CB = product of reflexions on CP and CI
- s is product of reflexions on CI and EP (orthogonal to BC)
- s*c is then product of reflexions on RC and RE
- draw on BP triangle BPQ, similar to triangle BFC. Its vertex Q moves on the bisector AF
- b*s*c is then the product of reflexions on QP and QB
- rotate triangle BQP about Q by the angle AQB to bring in the place XQY
- a*b*s*c becomes the product of reflexions on QY and AC
- QY and AC are orthogonal, thus defining a point-symmetry at their intersection K
- |JK| = |CE|. The equality of lengths is shown by calculating the dependence of |JK| from |CE|, or by showing that these lengths are linearly dependent and when E takes the position of G, then K coincides with C.
This nice result is used in the figure of the file Rotation_product_odd.html .

A similar result for quadrangles, that can be proved along the same line of arguments, is contained in the file Isometries_product2.html .

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