In a convex 1897-sided polygon distances from chosen point to all sides independent of the choice of the point

Question

Prove that in a convex 1897-sided polygon, in which all internal angles are equal, for any point inside this polygon, the sum of the distances from this point to all sides of the polygon is independent of the choice of the point. I tried to solve the problem geometrtically, but without succes.

Answer 1

This is true of any polygon in which all side lengths are equal. For each side the distance takes the form $(\vec z\times\vec s_i)\cdot\vec p+d_i$, where $\vec z$ is a vector orthogonal to the polygon's plane, $\vec s_i$ is a unit vector along the side, $\vec p$ is the vector for the point and $d_i$ is some constant independent of $\vec p$. Summing this over $i$ yields

$$ \sum_i\left((\vec z\times\vec s_i)\cdot\vec p-d_i\right)=\left(\vec z\times\sum_i\vec s_i\right)\cdot\vec p+\sum_id_i\;. $$

If all sides all have the same length $L$, then $\sum_i\vec s_i=\frac1L\sum_iL\vec s_i$ is a multiple of the sum of the vectors for all sides, which is zero since the sides form a closed polygon. Thus the sum of the distances is just the constant $\sum_id_i$.