A Diagonal Inequality

This inequality can be proved by realizing that the square roots in the inequality are the lengths of the diagonals of the four cuboids shown.
Specifically, three cuboids , and of dimensions , and fit inside the cuboid of dimensions corner to corner. Call the corners , , and . The diagonal of cuboid is , and the length of the broken line is the sum of the diagonals of , and , which is , due to the extended triangle inequality.


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[1] N. Stojanović, "Proof without Words: Inequality for the Sum of the Diagonals," The Mathematical Intelligencer, 43(4), 2021 p. 32. doi:10.1007/s00283-021-10075-9.
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