Ment
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I wrote that it's possible to translate the problem into a plane and prove that for any point inside a convex polygon, you can draw a perpendicular to one of the sides, so that the end of the perpendicular falls on the side itself, and not its extension.
Well, there you kind of divide this thing into triangles with angles vertex-vertex-center and calculate the heights, maybe it's also worth considering the 360 degrees and so on, I don't feel like solving it further.
Because, as Uranian pointed out, they would have used it a long time ago.