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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.

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

Geometry

Theory of Equations

Number Theory

But try the problem first...

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

Source

Suggested Reading

HANOI, 2018

Geometry Vol I to IV by Hall and Stevens

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 \((6\sqrt3)/4\)=\((3\sqrt3)/2\).

- https://www.cheenta.com/theory-of-equations-and-number-theory-aime-2009/
- https://www.youtube.com/channel/UCK2CP6HbbQ2V6gn8Xp_vM-Q

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