# How many equilateral triangles can be inscribed in a triangle?

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

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.
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