2022 AIME II Problems/Problem 11
Problem
Let be a convex quadrilateral with , , and such that the bisectors of acute angles and intersect at the midpoint of . Find the square of the area of .
Solution
According to the problem, we have , , , , and
Because is the midpoint of , we have , so:
Then, we can see that is an isosceles triangle with
Therefore, we could start our angle chasing: .
This is when we found that points , , , and are on a circle. Thus, . This is the time we found that .
Thus,
Point is the midpoint of , and . .
The area of this quadrilateral is the sum of areas of triangles:
Finally, the square of the area is
~DSAERF-CALMIT
See Also
2022 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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