Joseph sent the following, and I printed it.
Livio Zucca -- Joseph DeVincentis solution is not the best!!! The best solution is with 6 triangles! The first solver of my puzzle was Paolo Licheri on an Italian News Group. Here below a little correspondence with Michael Reid. "hi, thanks for your interesting dissection puzzle; i found a solution with 6 pieces. i assume you didn't have fewer than that. mike"
Solvers were baiocchi, Jeffrey Gosselin, and
Joseph DeVincentis -- I wouldn't have thought it possible, but if you connect all 6 vertices of the hexagon to a point (11-sqrt(21))/2 units from the left-most vertex along the horizontal axis toward the opposite vertex, all 6 resulting triangles are similar, in a ratio of approximately 3.209:4.005:5 as you go from left to right on either side of the horizontal axis.