UA-UMO 1997 Problem 97

Triangles $ABC$ and $A_1 B_1 C_1$ are non-congruent, but $AC=A_1 C_1=b,$ $BC=B_1 C_1=a$ , and $BH=B_1 H_1$ , where $BH$ and $B_1 H_1$ are the altitudes. Prove the inequality:

$a \cdot AB+b \cdot A_1 B_1 \le \sqrt{2}(a^2+b^2).$