Some years ago, I posted two geometrical problems related with side-ratios in triangles and squares. This year, I was on my bed thinking when I remember this two problems and I try to generalize for regular polygons of n-sides. After some calculations in Geogebra, I found an interesting relation between the regular polygon (the original one) and the builded polygon which was a linear equation. The question from which the problem emerges is:

Let be a polygon of n sides (original) with vertex labeled from \(A\) to \(N\). Between \(AB\) there is a point \(a\) such \(Aa=aB\), between \(BC\) there is a point \(b\) such \(Bb=\frac{1}{3}BC\), between \(CD\) there is a point \(c\) such \(Cc=\frac{1}{4}CD\) and so on until complete the n vertex (builded).

Then, polygons' areas are plotted (axis-\(x\) for the area of builded polygons and axis-\(y\) for the area of the original polygons) to values from \(n=1\) to \(n=17\)

A linear equation is got it from the previous question.

For polygons of side one I got the areas in te table below which its graph

I did the same process for polygons of sides \(2\) to \(7\) and get the same linear equation. I'm intrigued about:

Why this data shows a linear equation?

How can I get a matemathical process to calculate the builded area?

Why the linear equations is so close from \(y=x\)?

Those are my main questions, I hope someone get interested in this problem and together we could work together on this.

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