RU-SPB 2015 Problem 107

There is convex $n-$ gon. We color all its sides and also diagonals, that goes out from one vertex. So we have $2n-3$ colored segments. We write positive numbers on colored segments. In one move we can take quadrilateral $ABCD$ such, that $AC$ and all sides are colored, then remove $AC$ and color $BD$ with number $\frac{xz+yt}{w}$ , where $x,y,z,t,w$ - numbers on $AB,BC,CD,DA,AC$ . After some moves we found that all colored segments are same that was at beginning. Prove, that they have same number that was at beginning.