Triangles on planar Jordan $C^1$-curves
  We prove that a Jordan $\calc^1$-curve in the plane contains any non-flat triangle up to translation and homothety with positive ratio. This is false if the curve is not $C^1$. The proof uses a bit configuration spaces, differential and algebraic topology as well as the Schoenflies theorem. A partial generalization holds true in higher dimensions. 
Jean-Claude Hausmann
math.MG