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?
Solution 1 (Law of Cosines)
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 the 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 .
Note: We could’ve also noticed that the centroid divides the median into segments of ratio ~peelybonehead
Without loss of generality, let the centroid of be . Then, one of the vertices must be the other curve of the hyperbola. Without loss of generality, 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 .
Without loss of generality, 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 .
Without loss of generality, let the centroid of be . Assuming we don't know one vertex is we let the vertices be
Since the centroid coordinates are the average of the vertex coordinates, we have that and
We also know that the centroid is the orthocenter in an equilateral triangle, so Examining slopes, we simplify the equation to . From the equation we get that . These equations are starting to resemble Vieta's:
are the roots of the equation . This factors as for the points . The side length is clearly , so the square of the area is
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