几何作图题 凸五边形连接对角线形成一个五边形 这样迭代下去收敛到一个内点

hbghlyj

凸五边形$P=p_1p_2p_3p_4p_5$连接对角线形成一个五边形$T(P)=p_1'p_2'p_3'p_4'p_5'$
$P,T(P),T(T(P))\ldots$这样迭代下去，会收敛到$P$内部的一个点$c(P)$，见《The pentagram map》的Theorem 2.1.

问题：给定凸五边形$P=p_1p_2p_3p_4p_5$，点$c(P)$如何作图？

ejsoon

一個類似的問題是，三角形内接圓再内接正方形再内接正五邊形……求最終圓的半徑極限

hbghlyj

ejsoon 发表于 2024-12-30 09:10
一個類似的問題是，三角形内接圓再内接正方形再内接正五邊形……求最終圓的半徑極限 ...

If we inscribe a triangle inside a circle, another circle inside the triangle, then inside the circle we inscribe a square, and then another circle inside that square, etc. Then the equation relating the inradius and circumradius of a regular polygon gives the ratio of the final and initial circles as,
$$P =\displaystyle\prod_{n=3}^{\infty} \cos\left(\frac{\pi}{n}\right)$$
which is equal to the polygon inscription constant given in (OEIS A085365) as $K = 0.1149420448532962007\ldots$ . You can read more here. Hope this helped