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Problem of the Day #260: Cyclic Diamonds December 4, 2011

Posted by Albert in : potd , trackback

A circle of radius $1$ has diameter $\overline{AB}$ and chord $\overline{CD}$ perpendicular to $\overline{AB}$, intersecting $\overline{AB}$ at point $E$. If the maximum possible area of quadrilateral $ABCD$ is $M$, what is the smallest length of $AE$ so that the area of $ABCD = \frac{M}{n}$? Express your answer in terms of $n$.

Also, prove that for integer $n > 1$, length $AE$ is an irrational number.

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