# Geometric mean in geometry

Geometry Level 5

Large version of the image
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 $$XYZ$$.

×