Writing:
L = 1;
xc = 1;
yc = 1;
θ = Pi/6;
plot1 = ContourPlot[
Max[(x - xc) Sin[θ + 0 Pi/3] - (y - yc) Cos[θ + 0 Pi/3],
(x - xc) Sin[θ + 2 Pi/3] - (y - yc) Cos[θ + 2 Pi/3],
(x - xc) Sin[θ + 4 Pi/3] - (y - yc) Cos[θ + 4 Pi/3]] == L/(2 Sqrt[3]),
{x, 0, 2}, {y, 0, 2}, PlotPoints -> 50];
plot2 = Graphics[{EdgeForm[{Red, Thick}],
FaceForm[None],
Rotate[RegularPolygon[{xc, yc}, L/Sqrt[3], 3], θ]}];
Show[plot1, plot2]
I get:
and I just can not understand the reason why the two graphics do not coincide.