n边形与内接n边形的相似构造

几何画板及绘图内容加载中，请稍等……

2021-11-03 06:22

本范例演示在一个任意n边形内部构造一个与之相似的内接n边形。通过调整n的值（滑块控制），观察内外两个多边形的形态变化。核心数学概念包括相似变换、多边形性质及角度关系。作图要点：使用Sequence命令生成n边形的顶点序列，通过Polygon命令绘制多边形，利用Intersect命令求取关键交点，再用Segment命令连接相应顶点形成内接多边形。适合中学数学课程中学习多边形、相似形及几何作图的学生使用，也可用于探索n边形内接多边形的通用规律。

GeoGebra 命令及参数

Sequence((cos(((2 * k) * π / n)), sin(((2 * k) * π / n))), k, 0, n + 2) → P, {(1, 0), (0.9555728057861407, 0.2947551744109042), (0.8262387743159949, 0.5633200580636221), (0.6234898018587336, 0.7818314824680298), (0.365341024366395, 0.9308737486442042), (0.07473009358642417, 0.9972037971811801), (-0.22252093395631434, 0.9749279121818236), (-0.5000000000000002, 0.8660254037844386), (-0.7330518718298263, 0.6801727377709194), (-0.900968867902419, 0.43388373911755823), (-0.9888308262251285, 0.14904226617617428), (-0.9888308262251285, -0.14904226617617447), (-0.9009688679024191, -0.433883739117558), (-0.7330518718298262, -0.6801727377709195), (-0.4999999999999996, -0.8660254037844388), (-0.2225209339563146, -0.9749279121818236), (0.07473009358642436, -0.9972037971811801), (0.3653410243663954, -0.9308737486442041), (0.6234898018587334, -0.7818314824680299), (0.8262387743159949, -0.563320058063622), (0.9555728057861408, -0.2947551744109039), (1, 0), (0.9555728057861407, 0.29475517441090426), (0.8262387743159947, 0.5633200580636223)}
Polygon(Sequence[P(k), k, 1, n]) → V
Sequence(Segment[P(k), P(k + 3)], k, 1, n + 1) → S, {0.8677674782351162, 0.8677674782351162, 0.8677674782351164, 0.8677674782351162, 0.8677674782351164, 0.8677674782351161, 0.8677674782351162, 0.8677674782351161, 0.8677674782351161, 0.8677674782351162, 0.8677674782351161, 0.8677674782351166, 0.8677674782351162, 0.8677674782351162, 0.8677674782351161, 0.8677674782351161, 0.8677674782351162, 0.8677674782351161, 0.8677674782351164, 0.8677674782351161, 0.8677674782351161, NaN}
Intersect(S(1), S(2)) → Q1
Intersect(S(2), S(3)) → Q2
Polygon(Q1, Q2, n) → V1, f, q, i_1, k_1, m_1, p_1, r_1, t_1, c_1, d_1, f_2, g, h_2, j_2, l_2, n_2, q_2, A, B, C, D, E, h, F, G, H, T, W, A_1, C_1, E_1, G_1, J_1, i, K_1, M_1, O_1, Q_1, S_1, U_1, W_1, j, k, l, m, p
Polygon(Q1, Q2, n) → V1', r_2, g_3, h_3, i_3, j_3, k_3, l_3, m_3, n_3, p_3, q_3, s_2, r_3, s_3, t_3, a_3, b_3, Z_1, A_2, B_2, C_2, D_2, t_2, E_2, F_2, G_2, H_2, I_2, J_2, K_2, L_2, M_2, N_2, a_2, O_2, P_2, Q_2, R_2, S_2, T_2, U_2, b_2, c_2, d_2, e_2, f_3
Segment(I, (1, 0)) → s
Segment(I, J) → r

AI 绘图提示语参考

请绘制一个可调整边数的n边形，并构造一个与它相似的内接n边形。通过调整n的值，可以看到外层和内层多边形如何同步变化。关键对象包括外层多边形、由顶点连接形成的内部线段，以及通过这些线段交点确定的内层多边形。构造顺序是先生成顶点序列，再绘制外层多边形，然后连接顶点形成内部线段，最后利用交点绘制内层多边形。

以上提示语为根据原始绘图命令整理而来，仅供参考。