# Divide an equilateral triangle into three similar parts?

Is it possible to divide an equilateral triangle into three similar parts, in which two are identical but the third one is of different size?

• What do you mean by "parts"? Do you mean any shape at all, or a specific shape? If the shape doesn't matter, you can chop off two corners in the same way and get two identical pieces with a different-sized third piece. Commented Dec 4, 2017 at 5:18
• @JānisLazovskis You can do that, but the third piece won't be similar to the other two, will it? Commented Dec 4, 2017 at 5:21
• @TannerSwett right, I took "similar" in the non-technical sense. Seems more difficult than I thought at first. Commented Dec 4, 2017 at 5:28

Yes.
I think this can be tweaked, so the width of the blue shape is any multiple of the other two, greater than $\phi=(1+\sqrt{5})/2$. When the trapezium is divided, only the two remaining trapeziums have to be the same shape as the original. The bounding value is when the two remaining trapeziums start to overlap.