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Example of polygon. I wonder how to find a point within $n$-sides polygonal shape that makes all sub-triangle areas equal. In an attached picture, how to find point $C$ that makes area $A_1=A_2=A_3=A_4=A_5$?

Thanks.

Another picture for clarifying. Example 2

kmmm
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  • Sorry, just attached it. – kmmm May 01 '20 at 14:55
  • What is special about this shape so that such a point exists? – user May 01 '20 at 15:37
  • Nothing. I just show an example of 5-sides polyshape. It could be any number of sides but normally no more than 7 sides polygonal shape. – kmmm May 01 '20 at 15:52
  • But why should such a point exist for this shape? It does not exist even for a (general) quadrilateral. – user May 01 '20 at 20:40
  • Sorry for making un-clear post. Let's me start again. Pretending you've pentagon and of course you knew 5 vertices coordinates. I just wonder if I can find a point that divides 5 triangle area inside pentagon as shown in my attached file equally. – kmmm May 02 '20 at 05:07

1 Answers1

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It is relatively easy to demonstrate that in a triangle $ABC$ the locus of points $X$ such that $[AXB]=[AXC]$ is the median $AA'$ (where $A'$ is the midpoint of $BC$).

With this observation, the construction of the point in question in an arbitrary polygon is the following. For any triangle consisting of three adjacent vertices of the polygon $A_{i-1}A_iA_{i+1}$ construct the median $A_iA'_i$ (where $A'_i$ is the midpoint of $A_{i-1}A_{i+1}$). If all $n$ such "trimedians" intersect in a common point, this is the point in question. If on the other hand they do not intersect in a common point (what is rather a rule for $n>3$), such a point does not exist.

Update:

Below is an example of a pentagon where the point in question (denoted as $F$) do exists. $ABCD$ and $AFDE$ are parallelograms. It is seen that (different from the cases of triangle and quadrilateral) the point $F$ coincides neither with the area centroid $Z$ nor with the vertex centroid $V$.

enter image description here

user
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  • For polyshape with all sides are equal, a common point that I'm asking for is a "Centroid" of that polyshape. Am I right? But regarding your answer, such a point rather not exist for polyshape with n > 3, right? – kmmm May 02 '20 at 13:41
  • @kmmm A polyshape can have several centroids (such as vertex, edge, area centroids). Which one do you have in mind? – user May 02 '20 at 15:17
  • Oh! I didn't know that a polyshape can have several centroids. Please teach me. I created, let say, pentagon using Matlab. If all sides of pentagon are equal, the area centroid (using command "centroid" in Matlab) and its center (averaging vertices x and y coordinates) will be the same point, right? If its sides are not equal, its centroid and its center would not be the same point. Am I understand correctly? – kmmm May 02 '20 at 16:29
  • I also added another picture to clarify my question. – kmmm May 02 '20 at 16:52
  • @kmmm I do not believe that the equality of the pentagon sides suffice for coincidence of the vertex and area centroids. But it would be interesting if you give an example (distinct from the regular pentagon). – user May 02 '20 at 18:54
  • I don't know. That's why I'm asking question here. Did you see my Example 2? Point 4 on that picture obtained from "Centroid" command in Matlab. – kmmm May 03 '20 at 02:47
  • @kmmm I would assume that the "centroid" command produces area centroid aka gravity center. Whether it coincides in a $n$-gon with the point you are asking about, if the point exists, deserves a separate question. In general the point however does not exist. In the case of pentagon even no example (except for regular shape) is known where such point exists. But you can try using the recipe given in my answer to construct such an example. – user May 03 '20 at 06:42
  • That helps. Thanks for your helps. – kmmm May 03 '20 at 12:00
  • @kmmm I have found an example of a pentagon, where such a point exists. It does not coincide with the area and the vertex centroids. – user May 03 '20 at 17:39
  • Ok, but it exists for that particular pentagon, not for the general n-sided polygon. – kmmm May 09 '20 at 08:09
  • @kmmm Yes, of course. I added the example only to demonstrate that the point if it exists does not in general coincide with known centroids. – user May 09 '20 at 08:22