Maths geometry past year SAT question having a semicircle inside a triangle

Question

I tried very very hard to find the answer of this question but in vain. Can anyone pls show me how to solve this question?

My working:

If we draw perpendicular from $D$ to $BC$ and mark it $K$ then $BK = R$, if we draw a perpendicular from $E$ to $BC$ and mark the touching of $E$ perpendicular as $L$ then $KL = 2R$.

Answer 1

That seems promising. $LC= LE \cot 60° = \frac{R}{\sqrt 3}$ and so $$BC= R+2R +\frac{R}{\sqrt 3} $$

But on the other hand, $BC=26$. So you just need to solve $$R\left(3+\frac{1}{\sqrt 3} \right)= 26$$

Answer 2

Good. You're on the right track. Now consider triangles $\triangle DBK$ and $\triangle ECL$. They're both right triangles, but what are their other angles? Are they special triangles with memorable trigonometric ratios? Can you form an equation in $R$ for the length of $BC$?