2021 Fall AMC 12B Problems/Problem 9
Problem
Triangle is equilateral with side length . Suppose that is the center of the inscribed circle of this triangle. What is the area of the circle passing through , , and ?
Solution 1 (Cosine Rule)
Construct the circle that passes through , , and , centered at .
Also notice that and are the angle bisectors of angle and respectively. We then deduce .
Consider another point on Circle opposite to point .
As is an inscribed quadrilateral of Circle , .
Afterward, deduce that .
By the Cosine Rule, we have the equation: (where is the radius of circle )
The area is therefore .
~Wilhelm Z
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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