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

Geometry Level 5

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.

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