2022 AIME II Problems/Problem 15
Problem
Two externally tangent circles and have centers and , respectively. A third circle passing through and intersects at and and at and , as shown. Suppose that , , , and is a convex hexagon. Find the area of this hexagon.
Solution
First observe that and . Let points and be the reflections of and , respectively, about the perpendicular bisector of . Then quadrilaterals and are congruent, so hexagons and have the same area. Furthermore, triangles and are congruent, so and quadrilateral is an isosceles trapezoid. Next, remark that , so quadrilateral is also an isosceles trapezoid; in turn, , and similarly . Thus, Ptolmey's theorem on yields , whence . Let . The Law of Cosines on triangle yields and hence . Thus the distance between bases and is (in fact, is a triangle with a triangle removed), which implies the area of is .
Now let and ; the tangency of circles and implies . Furthermore, angles and are opposite angles in cyclic quadrilateral , which implies the measure of angle is . Therefore, the Law of Cosines applied to triangle yields
Thus , and so the area of triangle is .
Thus, the area of hexagon is .
~djmathman
See Also
2022 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Problem | |
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