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# Largest Hexagon in Equilateral Triangle | HANOI 2018

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

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

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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