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Given any triangle ABC find points D, E and F not A, B or C, where D is on segment AB, E on segment BC and F on segment CA, such that triangle DEF is equilateral. How many such triangles exist? I can construct at least 1. I feel but cannot prove that there are no more than 3. Please help.

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I think this is more reasonable question after the edit, and I'm reopening your question. –  mixedmath Apr 20 at 4:41
    
In general, you should expect infinitely-many inscribed equilaterals. Certainly, if $\triangle ABC$ is itself equilateral, then any symmetrically-placed $D$, $E$, $F$ give equilateral $\triangle DEF$. Otherwise, conditions $|DE|=|EF|$ and $|DE|=|DF|$ lead to two equations in three unknowns, say, $|AD|$, $|BE|$, $|CF|$. ($|EF|=|DF|$ gives a dependent equation.) Thus, one unknown is "free" and can "usually" take on infinitely-many values in a range. (If there are two inscribed equilaterals, say for $D=D_1$ and $D=D_2$, then there's an inscribed equilateral for any $D$ between $D_1$ and $D_2$.) –  Blue Apr 20 at 11:34

2 Answers 2

There's a reason you can't prove there are no more than three inscribed equilaterals:

enter image description here

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It depends. Do you mean without overlapping? Must the triangle be completely covered by the equilateral triangles? Must the triangles be lying on the triangle simultaneously?

A particular case is if the given triangle is equilateral. You can see in the figure, depending what inscribed means, that we get $4$ equilateral triangles.

enter image description here

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When "inscribed" means that all vertices of the inscribed triangle have to lie on the circumference of the given triangle, the answer is $\infty$: The given triangle has at least one angle $\leq60^\circ$, and it is then easy to inscribe small equilateral triangles with two vertices on one adjacent side and the third vertex on the other adjacent side. –  Christian Blatter Apr 18 at 18:24
    
if 'inscribed' means that the vertices of the inscribed triangle lie on the given triangle and none of the vertices are coincident with the vertices of the given triangle, then are you sure there are an infinite number? –  Reb Dave Apr 18 at 22:12

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