Mock AIME 2 2006-2007 Problems/Problem 12

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In quadrilateral $\displaystyle ABCD,$ $\displaystyle m \angle DAC= m\angle DBC$ and $\displaystyle \dfrac{\text{area} \triangle ADB}{\text{area} \triangle ABC}=\dfrac12.$ If $\displaystyle AD=4,$ $\displaystyle BC=6$, $\displaystyle BO=1,$ and the area of $\displaystyle ABCD$ is $\displaystyle \dfrac{a\sqrt{b}}{c},$ where $\displaystyle a,b,c$ are relatively prime positive integers, find $\displaystyle a+b+c.$

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AoPS users 4everwise and Altheman collaborated to create this problem.

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