How to draw more precise of iterations in this type of constructions?

Question

In this question I made this construction

Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $A_{n+1}B_{n+1}C_{n+1}D_{n+1}E_{n+1}$ as follows:

• $A_{n+1}$ is the intersection between the angle bisector of $\angle C_n $ and $\angle D_n$.
• $B_{n+1}$ is the intersection between the angle bisectors of $\angle D_n$ and $\angle E_n$.
• $C_{n+1}$ is the intersection between the angle bisectors of $\angle E_n$ and $\angle A_n$.
• $D_{n+1}$ is the intersection between the angle bisectors of $\angle A_n$ and $\angle B_n$.
• $E_{n+1}$ is the intersection between the angle bisectors of $\angle B_n$ and $\angle C_n$. (the two opposite angles)

The reason why I chose this construction is that the point $A_{n+1}$ is the only point that doesn't depend on $A_n$. I am allowing self-intersecting polygons in this constructions.

I drew the first 50 iteration using Geogebra here. However, zooming in on the 50th iteration resulted in significant lag and imprecise rendering, hindering further exploration. Additionally, manually coding the constructions in GeoGebra was time-consuming (over 6 hours for 50 iterations).

I am working on a general version of this question which need a lot of sequences to be drawn at once and using GeoGebra doesn't seem to be convenient.

Are there alternative tools better suited for precisely rendering a large number of iterations in this pentagon construction? While I suspect Python might be a viable option, my limited coding experience hinders exploration.

Answer 1

Instead of constructing the iterated points, you might consider calculating them.

For instance, one can show that, starting with pentagon $ABCDE$, the angle bisectors at $C$ and $D$ meet at a point expressible as a weighted average of vertices $B$, $C$, $D$, $E$:

$$A' := \frac{ \beta B + \gamma C + \delta D + \epsilon E }{ \beta+\gamma+\delta+\epsilon }$$ where $$\begin{align} \beta &= \phantom{-}|CD||CDE| \\[4pt] \gamma &= -|CD||DEB| + 2|DE||BCD|\\[4pt] \delta &= -|CD||BCE| + 2|BC||CDE| \\[4pt] \epsilon &= \phantom{-}|CD||BCD| \end{align}$$ and $|PQ|$ is the (unsigned) length of $\overline{PQ}$, while $|PQR|$ is the signed area of $\triangle PQR$ (positive or negative, according as $P\to Q\to R$ traces the triangle counter-clockwise or clockwise).

For the other derived points $B'$, $C'$, $D'$, $E'$, simply cycle the elements of the formula for $A'$ via $A\to B\to C\to D\to E\to A$.

Implementing this geometrically in GeoGebra could be tricky, since I don't think the software naturally understands signed area. I recommend running your experiments with numerical coordinates, letting the shoelace formula take care of signing areas automatically. (Besides, simply crunching numbers will be lightning-fast at every iteration.)

It's possible to continue symbolically, obtaining formulas for various lengths ($|A'B'|$, etc) and areas ($|A'B'C'|$, etc) within the new pentagon. Such formulas aren't super-terrible, but they're a bit messy to write down, so I won't bother, especially since they probably aren't any more useful for experimentation than just working with the numerical coordinates.

(If sufficiently-many numerical trials seem to confirm convergence, it might be worth taking a harder look at the symbolic representations to see how they respond to iteration. But that's a consideration for another time.)