Areas related with a linear equation

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 AA to NN. Between ABAB there is a point aa such Aa=aBAa=aB, between BCBC there is a point bb such Bb=13BCBb=\frac{1}{3}BC, between CDCD there is a point cc such Cc=14CDCc=\frac{1}{4}CD and so on until complete the n vertex (builded).

Then, polygons' areas are plotted (axis-xx for the area of builded polygons and axis-yy for the area of the original polygons) to values from n=1n=1 to n=17n=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 22 to 77 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=xy=x?

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

Note by Paola Ramírez
7 months, 3 weeks ago

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