The shortest path that meets every extended line that passes through a circle of radius $r$ Problem: 
A straight underground pipe passes by a statue at a maximum distance of 100 meters. The exact whereabouts of this pipe is unknown. What is the shortest trench i must dig that will assure me to find the arbitrary laid pipe.  

The answer appears to be a half circle with two lines of length 100 m aranged like a horse shoe magnet. 
If this is the shortest trench path how could such be proven formally. e.g disproving all other trench paths. 

the bold line represent the shortest trench path

is it worth investigating this problem for IB MATHS HL INTERNAL ASSESMENT?
Without typo, length is equal to $ 2 r + \pi r.$
Without typo, length is equal to $ 2 r + \pi r.$