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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