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).$
Source: Ukraine 1997 grade 9
Tags: inequalities, geometry proposed, geometry
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).$