Loci From Foci

Geometry Level 4



Two points \(F_1,F_2\) and a point \(P\) satisfy that \(|F_1P-F_2P|=2\) and \(F_1F_2=4\). Let \(I\) be the incenter of \(\triangle F_1PF_2\). As \(P\) varies, \(I\) traces out two disjoint loci. What is the sum of the lengths of these two loci? Round to the nearest thousandths.

\(\text{Details and Assumptions}\)

The incircle is drawn inaccurately on purpose.


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