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.

• I think this is more reasonable question after the edit, and I'm reopening your question. Apr 20 '14 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 '14 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. Apr 18 '14 at 18:24