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 $\boxed{\textbf{(B)$ (Error compiling LaTeX. Unknown error_msg)12\pi$}}$ (Error compiling LaTeX. Unknown error_msg).
~Wilhelm Z
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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