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Any elegant ideas?
I have a couple of ideas of how to prove this (will need to look into it further), but may I ask how you came to this point? Was it a question or did you arrive at this stage from another question? Do you have any ideas about how to prove this? Forgive the masses of questions, but I am curious.
Well, they would have to be separate proofs.
So to prove that it is impossible to divide the equilateral triangle into two equilateral triangles....I think it would be easier to do an indirect proof.
And there would be two cases:
Where there is a point D somewhere on the equilateral triangle ABC that is connected to one of the vertices (let's use A for this example) to form two equilateral triangles.
Where there is a point D and a point E somewhere on the equilateral triangle ABC that forms two equilateral triangles.
For the first case, assume that triangle ABD and triangle ACD are equilateral (it is given that ABC is equilateral). Therefore, mangle BAD = 60 degrees and mangle CAD = 60 degrees. Also, mangle BAD + mangle CAD = mangle BAC. The mangle BAC = 60 degrees, according to the definition of an equilateral triangle. So, transitively, mangle BAD + mangle CAD = 60. However, this is impossible. Therefore, the assumption must be false.
For the second case, assume that there are two points D and E that are not any of their vertices, and divide triangle ABC into two equilateral triangles. So in effect, it is a triangle plus the two points on the triangle. Since the two points could be any two points, we can just view the problem as a line that has to pass through the two sides of a triangle. There is no configuration where you can pass a line through the triangle without intersecting any of the vertices to get two triangles, so the assumption must be false.
Therefore, if case 1 and case 2 are false, then the statement must be false.
I'm not sure if case 2 is correct, but it was the only thing I could think of.
And I'm pretty sure proving that you can't have 3 and 5 equilateral triangles will take exponentially more time and a more complex proof, so I'm not going to bother trying that haha
Hope I could help