There are three possible orientations for the largest square inside an equilateral triangle, by symmetry.
There are also three possible orientations for the largest square inside a special isosceles triangle called the Calabi triangle. (see wikipedia).

 base of Calabi Triangle, assuming sides have length 1

In[1]:= base = x /. Solve[2 x^3 - 2 x^2 - 3 x + 2 == 0, x, Reals][[3]]

Out[1]= Root[2 - 3 #1 - 2 #1^2 + 2 #1^3 &, 3]

height of triangle 

In[2]:= height = y /. Solve[y^2 + (base/2)^2 == 1, y][[2]]

Out[2]= 1/2 Sqrt[4 - Root[2 - 3 #1 - 2 #1^2 + 2 #1^3 &, 3]^2]

 vertices of triangle

In[3]:= ctrip = Polygon[{{-base/2, 0}, {base/2, 0}, {0, height}}];

square centered at {xc,yc} with side a, rotated by angle \[Theta] 

In[4]:= sqr[{xc_, yc_}, a_, \[Theta]_] = 
  Polygon[(a/2 (RotationMatrix[\[Theta]].#) & /@ {{-1, -1}, {-1, 1}, {1, 
        1}, {1, -1}}) /. {x_, y_} -> {x + xc, y + yc}];

 Find the three orientations

In[5]:= sln1 = FindMaximum[{a, 
   RegionWithin[ctrip, sqr[{xc, yc}, a, \[Theta]]] }, {{a, .5}, {\[Theta], 
    0}, {xc, 0}, {yc, 0}}]

Out[5]= {0.448612, {a -> 0.448612, \[Theta] -> -5.33582*10^-21, xc -> -4.00508*10^-17,
   yc -> 0.224306}}

In[13]:= p1 = Show[Graphics @ {Opacity[0.3], Green, ctrip}, 
   Graphics @ {Opacity[0.3], Red, sqr[{xc, yc}, a, \[Theta]]} /. sln1[[2]], 
   ImageSize -> Small];

In[7]:= sln2 = FindMaximum[{a, 
   RegionWithin[ctrip, 
    sqr[{xc, yc}, a, \[Theta]]] }, {{a, .5}, {\[Theta], .6}, {xc, 0}, {yc, 
    0}}]

Out[7]= {0.448612, {a -> 0.448612, \[Theta] -> 0.682984, xc -> -0.0324308, 
  yc -> 0.315555}}

In[14]:= p2 = Show[Graphics @ {Opacity[0.3], Green, ctrip}, 
   Graphics @ {Opacity[0.3], Red, sqr[{xc, yc}, a, \[Theta]]} /. sln2[[2]], 
   ImageSize -> Small];

In[9]:= sln3 = FindMaximum[{a, 
   RegionWithin[ctrip, sqr[{xc, yc}, a, \[Theta]]] }, {{a, .5}, {\[Theta], 
    2 \[Pi] - .6}, {xc, 0}, {yc, 0}}]

Out[9]= {0.448612, {a -> 0.448612, \[Theta] -> 5.6002, xc -> 0.0324308, 
  yc -> 0.315555}}

In[15]:= p3 = Show[Graphics @ {Opacity[0.3], Green, ctrip}, 
   Graphics @ {Opacity[0.3], Red, sqr[{xc, yc}, a, \[Theta]]} /. sln3[[2]], 
   ImageSize -> Small];

 Check equality of square sizes

In[11]:= {sln1[[1]] - sln2[[1]], sln1[[1]] - sln3[[1]], sln2[[1]] - sln3[[1]]}

Out[11]= {-7.72961*10^-9, -7.72961*10^-9, 5.55112*10^-17}