2020 USAMTS Round 1 Problems/Problem 3
The bisectors of the internal angles of parallelogram with determine a quadrilateral with the same area as . Determine, with proof, the value of .
Solution 1
We claim the answer is Let be the new quadrilateral; that is, the quadrilateral determined by the internal bisectors of the angles of .
Lemma : is a rectangle. is a parallelogram. as bisects and bisects By the same logic, is a parallelogram. 2. and and By and we can conclude that is a rectangle.
Now, knowing is a rectangle, we can continue on.
Let and Thus, and By the same logic, and Because we have
Solution and by Sp3nc3r
Solution 2 (similar to Solution 1)
Let be the intersections of the bisectors of respectively.
Let . Then and . So, . Therefore, .
Similarly, .
So, therefore, must be a rectangle and
Now, note that . Also, .
So, we have
Since : for .
Therefore, by the Quadratic Formula, . Since , .
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