Triangle in a triangle: On a problem of Steinhaus

A necessary and sufficient condition on the sidesp, q, r of a trianglePQR and the sidesa, b, c of a triangleABC in order thatABC contains a congruent copy ofPQR is the following: At least one of the 18 inequalities obtained by cyclic permutation of {a, b, c} and arbitrary permutation of {itp, q, r} in the formula $$begin{array}{l} Max{ F(q^2 + r^2 - p^2 ), F'(b^2 + c^2 - a^2 )} + Max{ F(p^2 + r^2 - q^2 ), F'(a^2 + c^2 - b^2 )} le 2Fcr end{array}$$ is satisfied. In this formulaF andF′ denote the surface areas of the triangles, i.e. $$begin{array}{l} F = {textstyle{1 over 4}}(2a^2 b^2 + 2b^2 c^2 + 2c^2 a^2 - a^4 - b^4 - c^4 )^{1/2} F' = {textstyle{1 over 4}}(2p^2 q^2 + 2q^2 r^2 + 2r^2 p^2 - p^4 - q^4 - r^4 )^{1/2} . end{array}$$