We have a park that is triangle. We don't know the shape of the triangle and it can have any triangle shape and lengths. Where should we place a lamp to have light everywhere in the park?
my english is not enough to describe my answer. I think it's G (center of gravity, sorry again for bad English.). but I need to learn logic, why we should place a lamp to G or whatever answer is.
If you mean you want to maximize the amount of light at every point in the park. So you want to place the lamp in such a way that:
(1) you are equally far away from all three vertices, or
(2) you are at the midpoint between two vertices and closer to the third.
Thus (1) gives the circumcenter of the triangle and (2) gives the midpoint of the longest side. Can you work out what kinds of triangles use (1) and what kinds use (2)?
We'd have to make measurements to decide, but then we would always have to do that in order to place the lamp anyway.