On a geometrical problem connected with generalized billiards

We study a geometrical problem represented in two forms, which is connected with generalized newtonian billiards so that its solution is the trajectory of the corresponding billiard system. A special case of this problem is the classical problem on a determination of a shortest path between two points passing through a given line. It is obtained a full qualitative solution of the problem, and for countable number of cases it is shown that it is possible to find the solution by a construction with the help of compasses and ruler.