# Ew Ugly Diagram

Geometry Level 5

Let $$ABC$$ be a right triangle with right angle at $$C$$ such that $$AC = 3$$ and $$BC = 4$$. The circumcircle of $$ABC$$ has center $$D$$ and the incircle of $$ABC$$ touches the sides at points $$E, F,$$ and $$G$$, with $$E$$ on $$AB$$, $$F$$ on $$AC$$, and $$G$$ on $$BC$$. Triangles $$EDF$$ and $$FGD$$ have incenters $$H$$ and $$I$$, respectively. If $$HI = k$$, then find $$\lfloor 100 k \rfloor$$.

Details and assumptions:

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