The Triangle Inequality and Degenerate Triangles

A suspect for a robbery is brought in to be questioned. The suspect claims that he has been running some errands at the time of the robbery. This is what he said:

"I left home to get some groceries, the store is about half a mile away from my house. Then, I went directly from the supermarket to pick up some coffee beans from my favorite coffee shop, it is a bit more than half a mile walk there. After I picked up my coffee beans I walked home, it was almost $2$ miles! What a long walk! In fact, come to think of it, my house, the groceries and the coffee shop makes up a triangle!"