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 2 of 4: Reduce to a shortest broken line

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In plain words

After first reaching one vertex disk, the shortest continuation toward the other is straight.

|AX|+|XY|\ge |AX|+|XC|-r

Detailed analysis

Suppose the path first meets the disk centered at $B$ at $X$ , then meets the disk centered at $C$ . The final part is at least $XC-r$ , so the path length is at least $AX+XC-r$ . Straight segments attain this bound when $Y$ lies on $XC$ with $CY=r$ .