Try this beautiful problem from HANOI, 2018 based on Largest Hexagon in Equilateral Triangle.
Find the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side 3.
But try the problem first...
Answer: is \((3\sqrt3)/2\).
Geometry Vol I to IV by Hall and Stevens
Here suppose that the regular hexagon H with side a is inside the triangle equilateral triangle with side 3. Then, the inscribed circle of H is also inside the triangle, and its radius is equal to \((a\sqrt3)/2\)
On the other hand, the largest circle in the given equilateral triangle is its inscribed circle whose radius is \((\sqrt3/2)\).
It follows that \(a \leq 1\) and the answer is \((6\sqrt3)/4\)=\((3\sqrt3)/2\).