I found the answer to my question by using the algorithms suggested on [this website][1]. 

The idea is to find the along-cross distance defined as:

> **along-track distance**: distance between the start point and the closest point on the path to
> the third point

In simple words, it is the length of the great circle between the starting point (blue dot in my drwaing) and the generic projection along it (red dot in my drawing).

Since the goal is to find the coordinates (lat/lon) of the generic projected point along the great circle, we need to combine the following information:

 - Lat/lon of the starting point
 - Initial bearing angle
 - along-track distance

Such inputs allow to calculate the geographical coordinates of the projection along the great circle of a generically located point.

All the equations and algorithms are explained in the website previously linked.

  [1]: http://www.movable-type.co.uk/scripts/latlong.html