Geometry - Inadequate Information #2

Geometry Level 5

As shown in the figure above, a circle is inscribed in quadrilateral \(ABCD,\) where \(M\) and \(N\) are midpoints of \(\overline{AC}\) and \(\overline{BD},\) respectively, \(\overline{AB}=5, \overline{CD}=9\) and \(\overline{AC}^2+\overline{BD}^2+4 \overline{MN}^2=218.\)

Find the value of \(\overline{BC}\times\overline{DA}.\)

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