In the given figure, ABC is an equilateral triangle which is inscribed inside a circle and whose radius is r. Which of the following is the area of the triangle ?

In the given figure, ABC is an equilateral triangle which is inscribed inside a circle and whose radius is r. Which of the following is the area of the triangle ?
Correct Answer $${\left( {r - DE} \right)^{\frac{1}{2}}}{\left( {r + DE} \right)^{\frac{3}{2}}}$$

We have :
$$\eqalign{ & AE \bot BC{\text{ and }} \cr & AD = BD = CD = r \cr & AE = AD + DE = r + DE \cr} $$
In Δ BDC,
$$\eqalign{ & BE = \sqrt {{{\left( {BD} \right)}^2} - {{\left( {DE} \right)}^2}} \cr & \,\,\,\,\,\,\,\,\,\,\, = \sqrt {{r^2} - {{\left( {DE} \right)}^2}} \cr & \,\,\,\,\,\,\,\,\,\,\, = \sqrt {\left( {r - DE} \right)\left( {r + DE} \right)} \cr} $$
∴ Area of the triangle :
$$\eqalign{ & = \frac{1}{2} \times BC \times AE \cr & = \frac{1}{2} \times 2BE \times AE \cr & = BE \times AE \cr & = \sqrt {\left( {r - DE} \right)\left( {r + DE} \right)} \left( {r + DE} \right) \cr & = {\left( {r - DE} \right)^{\frac{1}{2}}}{\left( {r + DE} \right)^{\frac{3}{2}}} \cr} $$