Try this beautiful problem from HANOI, 2018 based on Largest Hexagon in Equilateral Triangle.

Geometry – HANOI 2018


Find the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side 3.

  • is \(3\sqrt7\)
  • is \((3\sqrt3)/2\)
  • is \(2\sqrt5\)
  • cannot be determined from the given information

Key Concepts


Geometry

Theory of Equations

Number Theory

Check the Answer


But try the problem first…

Answer: is \((3\sqrt3)/2\).

Source
Suggested Reading

HANOI, 2018

Geometry Vol I to IV by Hall and Stevens

Try with Hints


First hint

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 \((a\sqrt3)/2\)

Second Hint

On the other hand, the largest circle in the given equilateral triangle is its inscribed circle whose radius is \((\sqrt3/2)\).

Final Step

It follows that \(a \leq 1\) and the answer is \((6\sqrt3)/4\)=\((3\sqrt3)/2\).

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