PEDAL TETRAHEDRONS

Abstract

The concept of the pedal triangle of the given triangle with respect of the given point located in the plane of this triangle was introduced in the middle of the XX century. It was established that the third pedal triangle of the given triangle with respect of a given point in the plane of this triangle was similar to the initial. This are staid without study for a long time. The author of the present work identified some problems about pedal triangles. In particular, he proved the existence of some points in the plane of triangle such that their second pedal triangles with respect of these points were similar to initial. Then this problem was diffused onto the class of convex quadrangles and it was proved that fourth pedal quadrangle of the given quadrangle with respect of a given point in the plane of this quadrangle was similar to the initial. Besides, were studied the cases when second (third) pedal quadrangle of the given quadrangle with respect of a given point in the plane of this quadrangle was similar to the initial. The special case of the semicanonical trapezium was studied, some interesting geometric results were discovered here.

In the present work the next natural step of research is made, the concept of the pedal tetrahedron of the given tetrahedron with respect of a given point is introduced. The main result states that for any point of the space the fourth pedal tetrahedron of the given tetrahedron with respect of this point is similar to the initial. As an example the case of a right tetrahedron with canonical triangle in the base and pairwise congruent other three sides. It is proved that the altitude of this tetrahedron contains a point such that the first pedal tetrahedron with respect of this point is similar to initial.