I think I can do it with $4$ colors. I don't have a quick way to make a diagram, so I'll try to explain in words. (It would be easier if you had labelled the vertices.) I'll use the colors Red, Green, Blue and White.
Color one side of the outer triangle R, another side B, and the third side G. Color each side of the smaller triangles the same color as the parallel side of the big triangle. Now we color the $4$ segments on the line from each vertex of the big triangle to the center alternately W and X where X is the color of the side opposite this vertex. We start with W at the outer vertex and end with X at the center. Since X takes the values R,G,B, the edges that meet at the center are different colors.