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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