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:
Geogebra users, stay away. (Unless you're just harmlessly making a diagram for your own use, then you're welcome to answer.)
Once you find the exact answer, you may use a calculator.