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perimeter of triangle KLM on surface of a cube of length a > 2a\sqrt3

Source: 2002 Romania District VIII P4

May 24, 2020
perimetergeometry3D geometrycubegeometric inequality

Problem Statement

The cube ABCDA′B′C′D′ABCDA' B' C' D' has of length a. Consider the points K∈[AB],L∈[CC′],M∈[D′A′]K \in [AB], L \in [CC' ], M \in [D'A'].
a) Show that 3KL≥KB+BC+CL\sqrt3 KL \ge KB + BC + CL
b) Show that the perimeter of triangle KLMKLM is strictly greater than 2a32a\sqrt3.
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