My final solution required three such rotation/translations. First ABD is translated inside BDC. Then the smaller new triangle is translated twice more and you find an unexpected equilateral triangle.
The complexity here also puts this beyond the kids in math club's skill set. However, I think there's the start of an interesting exercise for the day present in these type of problems. Translation is not done much in school but is very simple to describe and then opens up lots of interesting areas. One of my favorite transforms is to take the two triangles created by the median of a triangle and then recombine them by sticking the divided edge together into a new but related triangle. These also can be thought of as tiling exercises. So I'm off to see if I can find a range of problems or activities that would be approachable.