point - Connecting centroids of lines sharing same intersection in QGIS?

I have this street network of the center of Turin (IT), composed by lines representing streets, and black points representing intersections.

I have computed the centroids of each line and displayed these as red points. Now I would like to connect the centroids (red points) of each line that share the same intersection (black points) as in the "B" scheme of this picture:

The attribute of the intersections, that are the black points, are simple Ids:

Whereas in the attributes of the lines representing the streets I have the Ids, the start node (that is the Id of the black point where the line begin) and the end node (the Id of the black point where the line end.)

Do someone have an idea about how this can be done?

I've thought about pharsing the lines start points over the end points and vice-versa, but I really don't know how to do this.