I am looking for the reference to the following theorem.
I have to apply a similar statement, and it would be nice to trace the source. Please note, I know few proofs in fact it is Problem 3 in my collection of exercises.
Theorem. Let $\gamma$ be a closed simple plane curve with curvature at most 1. Then the disc bounded by $\gamma$ contains a circle of radius $1$.
Here is an illustration for those who are lazy to read.