# Last Orthocenter Loses

Geometry Level 5

According to the diagram above (not drawn to scale), $$DG$$ is a chord on the circumcircle of $$ABC$$ such that $$DG$$ bisects $$BC$$ and $$AD\parallel BC$$. $$X$$ is a point on $$BC$$ satisfying $$XG\perp GD$$. Let $$H$$ denote the orthocenter of $$ABC$$.

Given $$XH=7, XG=13, h_b=23$$, where $$h_b$$ is the height from $$B$$ to $$AC$$. Find $$BH$$ if $$BH<20$$.

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