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 $A$ 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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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

Could you please upload a image of polygon ,say for n=4

vasu paliwal - 1 year, 4 months ago

Paola Ramirez posted this 1 year and 1 month ago.

A Former Brilliant Member - 1 year, 4 months ago

Yeah, I understand and I think she is not probably active on brilliant right now

I agree with you. vasu paliwal, when did you start solving problems in brilliant? Are you new here?

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$