A solution is given here (click on the second snapshot). The accompanying text says,
Dissecting a regular triangle into three similar polygons, two of them of the same size, looks quite impossible at first glance.
Hence it is no wonder that this problem was unsolved until 1987, when the author of this Demonstration [Karl Scherer] stumbled upon it as part of working on his book "A Puzzling Journey To The Reptiles And Related Animals" on irregular rep-tiles. The surprising solution can also be found in his Zilllions game 'Isolattice'. The solution became widely known when Martin Gardner published it.
No details are given for the Gardner publication.
Also worth a look is this paper, Shigeki Akiyama, Jun Luo, Ryotaro Okazaki, Wolfgang Steiner, Jörg Thuswaldner, Similar dissection of sets, which proves an equilateral triangle can be dissected in three similar copies whose areas have ratio $1:1:a$ for $a\ge(3+\sqrt5)/2$. But I think the solution is in fractals, not polygons. This paper also gives the Gardner citation.