Dilated Steiner Inellipse

Geometry Level 5

Triangle has vertices \(A=(0,0), B=(9,0), C=(3,6)\). Its Steiner inellipse is dilated from triangle centroid. New ellipse trisects all three sides of the triangle \(ABC\).

Given that the dilation factor can be expressed as \(\dfrac{a}{\sqrt{b}}\), where \(a\) and \(b\) are positive integers with \(b\) square-free.

Find \(a+b\).

The problem is original

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