On the existence of Steinhaus points. Elementary analysis

The paper discusses issues related to the well-known 'square' Steinghaus hypothesis, which has already hit the list of Richard Guy's unsolved problems three times. Consideration is carried out exclusively by elementary methods. That is, only those that could be used in working with schoolchildren on mathematical projects of the International Computer Schools for the Young (ICSS - 4–7, 13–15). The particular results obtained are very extensive: both in the winter and summer sessions of the ICJJ, the task was to obtain the most general (using elementary methods) results with minimal assumptions: we study a system of four Pythagorean equations with seven unknowns.
The numbering of formulas and lemmas is by paragraphs; the numbering of picturies and theorems is through. The designations of the variables in the main text are consistent with those used in Appendix 1 and the drawings cited therein. Variable notations and drawings in Appendices 2 and 3 follow Steinhaus's book Tasks and Thoughts.