# Interesting Parallelogram

Geometry Level 4

A point $$X$$ is drawn on the diagonal of a parallelogram $$ABCD$$. The lines parallel to the sides through $$X$$ are constructed, and quadrilaterals $$P$$ and $$Q$$ are formed, as shown in the diagram.

Given that $$Q$$ has base $$12$$ and perpendicular height $$4$$ and that the area of $$ABCD$$ is $$200$$, find the sum of all possible values of the base of $$P$$.

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