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 3 of 4: Minimize the broken line

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In plain words

The extremal contact point balances the two distances from the starting and final vertices.

X=(-1/8,\sqrt3/8), AX=XC=\sqrt7/4

Detailed analysis

With $A=(0,\sqrt3/2)$ , $B=(-1/2,0)$ , and $C=(1/2,0)$ , minimize $AX+XC$ subject to $|XB|=r$ . Reflection or the stationarity condition gives $X=(-1/8,\sqrt3/8)$ ; direct substitution gives $AX=XC=\sqrt7/4$ .