2005 Alabama ARML TST Problems/Problem 3
The difference between the areas of the circumcircle and incircle of an equilateral triangle is square units. Find the number of units in the length of a side of the triangle.
Let be the radius of the circumcircle and let be the radius of the incircle. Then we have . If the center of these two circles is , the vertices are and , and is the midpoint of side , triangle is a right triangle, and its hypotenuse has length and its shorter leg has length . Thus . (There are many other arguments to get to this conclusion; for instance, is also the centroid of the triangle and is a median, so trisects and .)
Then so and the side length of the triangle is equal to .
|2005 Alabama ARML TST (Problems)|
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