The steps to construct the original triangle(triangle BDE above) given its triangle of medians(the shaded triangle above) are as follows.
1. Draw the midpoints on sides BC and AC.
2. Construct a line threw the midpoint of side BC and point A.
3. Construct a line threw the midpoint of side AC and point B.
4. Construct a point(G) at the intersection of these two lines just drawn.
5. Construct a line through point G that is parallel to segment AC.
6. Construct a point on segment AB where this parallel line just crossed.
7. Through this point just drawn, construct a parallel line through it that is parallel to segment BC.
8. Place a point(E) at the intersection of this last line drawn and the line that passes through point B and the midpoint of AC.
9. Construct a line through points E and A.
10. Construct a line through point B that is parallel to the line passing through points AG.
11. Place a point(D) at the point of intersection between this line just drawn an the line that passes through points AE.
12. The original triangle defined by the given triangle of medians is now complete and it is the triangle BDE.
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See an animation of the Original Triangle given the Triangle of Medians