2017 AMC 10B Problems/Problem 24
The vertices of an equilateral triangle lie on the hyperbola , and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?
WLOG, let the centroid of be . The centroid of an equilateral triangle is the same as the circumcenter. It follows that the circumcircle must intersect the graph exactly three times. Therefore, , so , so since is isosceles and , then by Law of Cosines, . Alternatively, we can use the fact that the circumradius of an equilateral triangle is equal to . Therefore, the area of the triangle is , so the square of the area of the triangle is .
WLOG, let the centroid of be . Then, one of the vertices must be the other curve of the hyperbola. WLOG, let . Then, point must be the reflection of across the line , so let and , where . Because is the centroid, the average of the -coordinates of the vertices of the triangle is . So we know that . Multiplying by and solving gives us . So and . So , and finding the square of the area gives us .
WLOG, let the centroid of be and let point be . It is known that the centroid is equidistant from the three vertices of . Because we have the coordinates of both and , we know that the distance from to any vertice of is . Therefore, . It follows that from , where and , using the formula for the area of a triangle with sine . Because and are congruent to , they also have an area of . Therefore, . Squaring that gives us the answer of .
Solution 4 (5-second sol)
WLOG let the centroid of the triangle be . By symmetry, one of the vertices is . The distance between these two points is , so the height of the triangle is , the side length is , and the area is , yielding an answer of .
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