Constructing an equilateral triangle of a given side length inscribed in a given triangle 
I am trying to solve the problem of constructing, with straightedge and compass, an equilateral triangle of given side length $a$ inscribed in a given triangle.

I found this post "Inscribe an equilateral triangle inside a triangle" and this other post "How to draw an equilateral triangle inscribed in another triangle?" but the construction must be made with straightedge and compass, using simple constructions such as arcs, parallel lines, perpendicular lines and that kind of thing. 
I tried constructing the arcs capable of $120^{\circ}$ on the sides of the given triangles and noticed that the centers of the arcs form an equilateral triangle, but I don't know what to do after that.

I believe the following diagrams and incorporated explanation will suffice. Let me know if it is not clear. Click on image to get a larger and clearer view.

