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Journal of Computer-Generated Euclidean Geometry |
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Compositions of Transformations in Triangle Geometry Deko Dekov
Given a point P. Then Cyclocevian conjugate of P = Isotomic conjugate of the anticomplement of the isogonal conjugate of the complement of the isotomic conjugate of P. Below by transformation we mean one of the
following 5 transformations (with relation to a given triangle): Complement,
Anticomplement, Isogonal Conjugate, Isotomic Conjugate, Cyclocevian
Conjugate. By composition of one transformation we mean one of these
transformations. Trivial relation is one of the following 5 relations: Complement of the Anticomplement of P = P. If we order to a computer to give us all
non-trivial relations between compositions of transformations, except for
trivial relations, such that the number of all transformations in any
composition is no more than 3, we obtain the following 6 relations (not
included in the first edition of [2]): Isogonal Conjugate of Complement of Isotomic
Conjugate of Point P = Complement of Isotomic Conjugate of Cyclocevian
Conjugate of Point P. Isotomic Conjugate of Anticomplement of Isogonal
Conjugate of Point P = Cyclocevian Conjugate of Isotomic Conjugate of
Anticomplement of Point P. Anticomplement of Isogonal Conjugate of
Complement of Point P = Isotomic Conjugate of Cyclocevian Conjugate of
Isotomic Conjugate of Point P. Complement of Isotomic Conjugate of Cyclocevian
Conjugate of Point P = Isogonal Conjugate of Complement of Isotomic Conjugate
of Point P. Cyclocevian Conjugate of Isotomic Conjugate of
Anticomplement of Point P = Isotomic Conjugate of Anticomplement of Isogonal
Conjugate of Point P. Isotomic Conjugate of Cyclocevian Conjugate of
Isotomic Conjugate of Point P = Anticomplement of Isogonal Conjugate of
Complement of Point P. It is easy to see that any of the above 6
relations can be rewritten to the Grinberg's relation. We can make the
following conclusion: If we consider relations between compositions having
together no more than 6 transformations, we will not obtain new relations.
Indeed, any relation of the form composition of two transformations = composition
of 4 transformations rewrites to relation of the form composition of 3 transformations = composition
of 3 transformations, so that any relation between compositions having
together 6 transformations rewrites to Grinberg's relation. In addition, since in the general case there is
no relations between compositions having together 2, 3, 4 and 5
transformations, we can conclude that in the general case, a composition of
one or two or three transformations of P does not coincide with any
composition of one or two transformations of P. E.g., in the general case,
all points - compositions having no more than 2 transformations of a given
point, are different. Clearly, there are 25 non-trivial compositions of no
more than two transformations, e.g. Complement of P, Complement of the
Complement of P, Isogonal Conjugate of Complement of P, etc. Clearly, in special cases the points-images can
coincide, e.g. Complement of the Orthocenter = Isogonal Conjugate of the
Orthocenter, and so on. In [1], Grinberg introduced the following
composition of transformations: Isotomic Complement of P = Complement of the
Isotomic Conjugate of P. Additional compositions of transformations are
defined in [2]. The above remarks give us the basis to define additional
compositions of transformations. I would like to
note the usefulness of compositions of transformations in Triangle Geometry.
By using compositions of transformations, we are able to find useful relations
between points, triangles, circles. In many cases these relations give us
convenient ways we to construct new points by using straightedge and compass.
References
Publication Date: Dr.Deko Dekov, ddekov@dekovsoft.com |