2022 AIME I Problems/Problem 8
Contents
Problem
Equilateral triangle is inscribed in circle with radius Circle is tangent to sides and and is internally tangent to . Circles and are defined analogously. Circles , , and meet in six pointstwo points for each pair of circles. The three intersection points closest to the vertices of are the vertices of a large equilateral triangle in the interior of , and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of . The side length of the smaller equilateral triangle can be written as , where and are positive integers. Find .
Diagram
Solution 1
We can extend and to and respectively such that circle is the incircle of .
Since the diameter of the circle is the height of this triangle, the height of this triangle is . We can use inradius or equilateral triangle properties to get the inradius of this triangle is (The incenter is also a centroid in an equilateral triangle, and the distance from a side to the centroid is a third of the height). Therefore, the radius of each of the smaller circles is .
Let be the center of the largest circle. We will set up a coordinate system with as the origin. The center of will be at because it is directly beneath and is the length of the larger radius minus the smaller radius, or . By rotating this point around , we get the center of . This means that the magnitude of vector is and is at a degree angle from the horizontal. Therefore, the coordinates of this point are and by symmetry the coordinates of the center of is .
Solution 2
Let bottom left point as the origin, the radius of each circle is , note that three centers for circles are
It is not hard to find that one intersection point lies on since the intersection must lie on the angle bisector of the bigger triangle, plug it into equation , getting that , the length is , leads to the answer
~bluesoul
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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