Problem 4

A soldier must investigate mines in an equilateral triangular region. His detector has radius equal to one-half the triangle’s altitude, and he starts at one vertex. Determine the shortest path that checks the whole region.

Step 1 of 4: Normalize the geometry

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In plain words

The detector radius is fixed by the altitude, so scaling lets us set the side to one.

s=1, h=\sqrt3/2, r=h/2=\sqrt3/4

Detailed analysis

Let $A$ be the starting vertex and $B,C$ the others. Any successful path must come within $r$ of both $B$ and $C$ , because the detector must check those vertices.