I think that in order to know how to build a triangle given its medians, I must know some relationships
between the triangle and its medians triangle.
So, I first started building a triangle, any triangle, with its medians:

I noticed in GSP that when I moved any of the vertices of triangle ABC around, the following ratios always stay constant:

I then proceeded to finding a proof for this. Luckily for me, I found that there is a theorem in my old geometry book
which guarantees that point G (the centroid) is  2/3  of the distances from each vertex of  triangle ABC to the midpoint of the opposite side. I then used this fact to help me in the construction..

I constructed a triangle with medians as its sides:

In the figure, triangle FHC is the medians triangle. I then noticed that medians AE and BD were translated to line FH  and HC which
are parallel to line AE and BD respectively. They are pallerel to these lines because of the definition of translation. I then wondered what if I construct the medians triangle first then retrace the steps of what I just did above going backward from the medians triangle back to the original triangle ABC. I discovered that it is possible to do this.

I arbitrarily chose three segments  k, l, and m as the given medians of the triangle that I want to construct.

I then constructed the triangle having sengments k,l, and m as its three sides:

Triangle FCH is the medians triangle. I next trisected all thee sides FC, CH, and FH. I did this on purpose because I know from my doing exploration of the relationship between a triangle and its medians that the intersection point of the three medians is about 2/3 from each vertex of the triangle we intend to construct and because these points are keys points in the construction.

First, I trisected segments  FC, CH, and FH:

Then I drew a line paralellel to side CH through point F'. I then copied side CH onto this line and glide-translated it into a position such that it intersects CF at a point 2/3 in distances from one of its endpoints. That is, as you will see in the figure below, point C' of side CH was glided along parallel line q until it coincides with point F' of segment CF (note: line q is parallel to the lines above and below it)

                            
Now, it is a matter of gliding down side FH into the correct positional configuration which makes their intersection point (FC,CH,FH)
the centroid of the triangle I want to construct.

In the figure above, side FH is translated into position by gliding it along vector H"F'. Segment AF' is the same as segment FH but at a new position. A, B, and C are the vertices of the new triangle that I want to construct.

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