After finding this interesting question, I was wondering how could be proved the existence of some point $P$ inside any convex polygon $C=\{v_1,v_2,\dots,v_n\}$, where $v_k$ is the $k_{th}$ vertex and all the vertices are consecutive, such that, denoting $A_C$ the polygon's area, and $A_{v_iPv_j}$ the area formed by the triangle $v_iPv_j$, $$A_{v_iPv_j}=\frac{A_C}{n}\space\forall j=i+1$$
Any hint regarding this demonstration would be welcomed. Furthermore, if this point has some particular name, happy to hear about it and being redirected to the relevant documentation.
Thanks!
How about an equiangular hexagon whose edge lengths (in cyclic order) are $a, b, a, b, a, b$? When $a \gg b$ it is easy to see that no such point exists: the putative point must be equidistant from the three long sides, so would have to be the center of the hexagon. (Edited following comments to make a more complete argument.)