Construction Is A Piece Of Cake! Right? (Part-3)

Mohit, influenced by Akshat and Anshuman, took a right \(\bigtriangleup ABC\) (right angled at \(B\)), where \(AB=5.5 \mathrm{ cm}\) and \(BC=8 \mathrm{ cm}\), and started doing aimless constructions, the steps of which are given below:

(\(1\)) He drew perpendicular bisector \(X_1Y_1\) of \(AC\) which intersects \(AC\) at a point \(O,\) and another perpendicular bisector \(X_2Y_2\) of \(AB\) which intersects \(AB\) at a point \(G\).

(\(2\)) Then he constructed a circle taking center \(O\) and radius of the circle as \(OA\) . Then he constructed an \(\angle ACE\) which is equal to \(\angle ACB\) such that \(E\) lies on the circle.

(\(3\)) He then joined \(BE\) which meets \(AC\) at \(F\).

(\(4\)) Then he joined \(CG\) which intersects \(EB\) at \(H\) and \(X_1Y_1\) at \(J\).

Mohit then wondered what \(\frac{BH}{OJ}\) could possibly be equal to.