The remarkable properties of the discrete brachistochrone

The Brachistochrone (= shortest time) problem was one of the most famous problems in the 17th and in the 18th centuries. The question is:  what is the shape of the curve which a bead sliding from rest and accelerated  by gravity will slip frictionlessly from one point to another in the least time.
The solution to this problem was introduced by Johan Bernoulli in 1696 and later on it was solved also by Leibniz and Newton. Nevertheless, until recently the problem of a discrete Brachistochrone, in which the curve is made of straight segments, has never been solved.
In this lecture I will show three remarkable features of discrete Brachistochrone:
1. The sliding times on each segment are the same.
2. There is a constant difference between adjacent angles of the slopes of the segments.
3. There is a simple relation between the lengths of the segments.
Based upon these properties, one can show, in a simple and rigorous way, that in the limit where the number of segments tends to infinity, the shape of the discrete path merged with the curve of the cycloid which is the well-known solution of the original problem of the continuous Brachistochrone.