Proportional segments, a polygon, and Chevy's lines into a circle of the 7-th year students' research competence

It is received an equality analogous to Chevy's equality for a polygon inscribed into a circumference over the ends of the chords with cross point. A new version of proof of Ponsel's theorem is offered for a polygon inscribed into a circumference with an odd or even numbers of sides. It is offered an analog to Menelaus' theorem for a polygon of “star” type inscribed into a circumference.