Question 1149199
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Draw the radii of the circle to the three vertices of the equilateral triangle.<br>
Segment OA, the segment from A to the vertex at one end of the side of the triangle containing A, and the radius of the circle to that same vertex, form a 30-60-90 right triangle.<br>
Given leg OA = 2*sqrt(3), the other leg has length (2*sqrt(3))*sqrt(3) = 6.<br>
The altitudes of an equilateral triangle divide each other in the ratio 2:1; that means OA is 1/3 the length of the altitude of the equilateral triangle.  Given OA=2*sqrt(3), the altitude of the equilateral triangle is then 6*sqrt(3).<br>
The area of the equilateral triangle is one-half base times height; that is<br>
{{{(6)(6*sqrt(3)) = 36*sqrt(3)}}}<br>