2020 CIME I Problems/Problem 5

Problem 5

Let $ABCD$ be a rectangle with sides $AB>BC$ and let $E$ be the reflection of $A$ over $\overline{BD}$. If $EC=AD$ and the area of $ECBD$ is $144$, find the area of $ABCD$.

Solution


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Let $O$ be the center of rectangle $ABCD$. Because $E$ is the reflection of $A$ over $\overline{BD}$ and $\angle BAD = 90$ degrees, we have $\angle BED = 90$ degrees. This means $E$ lies on the circle with diameter $\overline{BD}$, or the circumcircle of rectangle $ABCD$. We are given $EC=BC$, so by symmetry, $EC=ED$. Since the three lengths are equal and $\overarc{BD}=180$ degrees, we must have $\overarc{BC}=\overarc{CE}=\overarc{ED}=60$ degrees, so $\triangle OBC$, $\triangle OCE$, $\triangle OED$ are all equilateral. Given that the area of cyclic quadrilateral $ECBD$ is $144$, the area of $\triangle OBC$ is $\frac{1}{3} \cdot 144 = 48$. This is $\frac{1}{4}$ of the area of rectangle $ABCD$, so the answer is $48 \cdot 4 = \boxed{192}$.

See also

2020 CIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All CIME Problems and Solutions

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