extra discussion

it will be interest if we think in a physical way

you have a cycle (D=10) and a long stick

you want to find a shortest string to-

1- surround the cycle with that string and the stick
2- the string must end at the stick (where you finish you trip)
3- the string starts at the center of the cycle
4- but in the middle somewere, the string must also reach the stick.

tighten the string, you will get the path of the shortest one.