3 Cheers for Geometry !!

Geometry Level 5

The sides a,b,c of a \( \Delta \)ABC are in Geometric Progression whose common ratio is \( \frac{2}{3} \) and the circumradius of the triangle is \( 6 \times \sqrt{ \frac{7}{209}} \) .

Find the longest side of the triangle .

The answer is of the form \( \frac{m}{n} \) where m and n are co-prime numbers ;

Find \( m^{n} \) .

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