ABOUT PEDAL QUADRANGLES

Abstract

․ In the second half of the previous century the concept of the pedal triangle with respect of the interior point for the given triangle was introduced (Alexander Oppenheim). It was established that the third pedal triangle with respect of an arbitrary point is similar to the initial triangle. It was the unique result about pedal triangles. We studied in details the geometry of different pedal triangles. For example, it was proved the existence of the interior point such that the second pedal triangle with respect of the same point is similar to the initial triangle. In the present stage of this research, results about pedal triangle are disseminating for the case of quadrangles. In the present article, we prove the analogue of the Oppenheim’s theorem for quadrangles: for any interior point of the convex quadrangle, the fourth pedal quadrangle with respect of this point is similar to the initial quadrangle. Besides, it is proved that if the sum of neighbor angles measures of the convex quadrangle is 180°, then the second pedal quadrangle with respect of the point of intersection of its diagonals is similar to the initial quadrangle. The results obtained will be of interest to high school students interested in geometry, as well as teachers and other mathematics specialists