Geometric mean in geometry

Geometry Level 5

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For \(\triangle ABC\), \(I\) is the midpoint of \(BC\), point \(D\) is in line segment \(AC\) such that \(CD=3AD\) and point \(E\) is in line segment \(AB\) such that \([BIE]=\sqrt{[CID] \times [ADE]}\).

If the ratio \(\dfrac{AE}{EB}\) can be expressed as \[\dfrac{\sqrt{a}-b}{c}\], then find \( a + b + c. \)

(The figure is not drawn to scale)

Note: \( [XYZ] \) denotes the area of triangle \(XYZ\).

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