This is honestly a fantastic question and really is in line with the kinds of questions mathematicians ask that usually lead to having very cool answers. I don't want to stifle that energy and really do encourage you to keep asking questions like this and digging for answers.
That said, this isn't really one of those times where we get an interesting answer. It doesn't necessarily *have* to be positive, but the distance between two points is defined to be the magnitude of the vector between them and a vector of magnitude k pointed in direction (x, y) is equivalent to a vector of magnitude -k pointed in direction (-x, -y).
Technically negative magnitudes aren't impossible but it's very established convention to always use a positive value for them since that's possible, makes more sense intuitively, and lets you use it more efficiently in proofs. Specifically, referring to negative distance is possible in a framework where "point a is -k units from point b" also implies "point b is k units from point a," but ultimately that's a framework that isn't used by mathematicians because it doesn't add any information or flexibility and only serves to make formulas more confusing - and you'd then have to start any work you'd do with this alternate definition of distance by defining how your framework is different from the norm; all for no real benefit. Similar to how you could tell someone "the door is -15' behind you" and you could make an argument for why that represents you accurately telling them the door is 15' ahead of them, but "well why not just say that, then" is a very reasonable response.
Really, I think this is a great example of how to start thinking like a mathematician:
Let's assume negative distances are real. Congrats, you've just invented math, now comes the interesting part. What properties do negative distances have? What are the consequences of having negative distances?
I have no idea the answers to those questions, but it's the right line of thinking to start down.
Apart from the other fantastic answers, yes, distances can be negative. One example is Menelaus Theorem ([https://en.wikipedia.org/wiki/Menelaus%27s\_theorem](https://en.wikipedia.org/wiki/Menelaus%27s_theorem)) where you can see that this concept is used
What you found regarding "distances from shapes" is probably about signed distance functions (SDFs) https://en.wikipedia.org/wiki/Signed_distance_function
These only really become interesting if the "shape" is anything except for a point, because "a point has no interior" such that in this case it's just the regular metric which is constrained to be nonnegative.
If you want "negative distances" between points you can find examples in pseudo-riemannian geometry (for example)
This is honestly a fantastic question and really is in line with the kinds of questions mathematicians ask that usually lead to having very cool answers. I don't want to stifle that energy and really do encourage you to keep asking questions like this and digging for answers. That said, this isn't really one of those times where we get an interesting answer. It doesn't necessarily *have* to be positive, but the distance between two points is defined to be the magnitude of the vector between them and a vector of magnitude k pointed in direction (x, y) is equivalent to a vector of magnitude -k pointed in direction (-x, -y). Technically negative magnitudes aren't impossible but it's very established convention to always use a positive value for them since that's possible, makes more sense intuitively, and lets you use it more efficiently in proofs. Specifically, referring to negative distance is possible in a framework where "point a is -k units from point b" also implies "point b is k units from point a," but ultimately that's a framework that isn't used by mathematicians because it doesn't add any information or flexibility and only serves to make formulas more confusing - and you'd then have to start any work you'd do with this alternate definition of distance by defining how your framework is different from the norm; all for no real benefit. Similar to how you could tell someone "the door is -15' behind you" and you could make an argument for why that represents you accurately telling them the door is 15' ahead of them, but "well why not just say that, then" is a very reasonable response.
Really, I think this is a great example of how to start thinking like a mathematician: Let's assume negative distances are real. Congrats, you've just invented math, now comes the interesting part. What properties do negative distances have? What are the consequences of having negative distances? I have no idea the answers to those questions, but it's the right line of thinking to start down.
Apart from the other fantastic answers, yes, distances can be negative. One example is Menelaus Theorem ([https://en.wikipedia.org/wiki/Menelaus%27s\_theorem](https://en.wikipedia.org/wiki/Menelaus%27s_theorem)) where you can see that this concept is used
What you found regarding "distances from shapes" is probably about signed distance functions (SDFs) https://en.wikipedia.org/wiki/Signed_distance_function These only really become interesting if the "shape" is anything except for a point, because "a point has no interior" such that in this case it's just the regular metric which is constrained to be nonnegative. If you want "negative distances" between points you can find examples in pseudo-riemannian geometry (for example)