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Inscribed Regular Polygons in a Circle

Inscribed Regular Polygons in a Circle
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2015-10-01 03:22
CirclePolygonAngleGeometric Construction
This example demonstrates how to inscribe regular polygons within a circle and explores the relationships between central angles, interior angles, exterior angles, and base angles of isosceles triangles formed. Key math concepts include circles, polygons, angle relationships, and geometric constructions. GeoGebra commands like Circle, Rotate, Polygon, and Angle are used in the construction. Suitable for high school students to understand properties of regular polygons and geometric transformations, as well as for mathematical teaching demonstrations.

GeoGebra Commands & Parameters

  1. Intersect(yAxis, xAxis) → A
  2. Circle(A, radius) → c
  3. Rotate(B, a, A) → B'
  4. Polygon(B, B', n) → poly1, b, l, k_5, l_5, m_5, n_5, p_5, q_5, r_5, s_5, t_5, a_6, m, b_6, c_6, d_6, e_6, f_6, g_6, h_6, i_6, j_6, k_6, p, l_6, m_6, n_6, p_6, q_6, r_6, s_6, t_6, a_7, b_7, q, c_7, d_7, e_7, f_7, g_7, h_7, i_7, j_7, k_7, l_7, r, m_7, n_7, p_7, q_7, r_7, s_7, t_7, a_8, b_8, c_8, s, d_8, e_8, f_8, g_8, h_8, i_8, j_8, k_8, l_8, m_8, t, n_8, p_8, q_8, r_8, s_8, t_8, a_9, b_9, c_9, d_9, a_1, e_9, f_9, g_9, h_9, i_9, j_9, k_9, l_9, m_9, n_9, b_1, p_9, q_9, r_9, s_9, t_9, a_{10}, b_{10}, c_{10}, d_{10}, e_{10}, c_1, f_{10}, g_{10}, h_{10}, i_{10}, j_{10}, k_{10}, l_{10}, m_{10}, n_{10}, p_{10}, d, d_1, q_{10}, r_{10}, s_{10}, t_{10}, a_{11}, b_{11}, c_{11}, d_{11}, e_{11}, f_{11}, e_1, g_{11}, h_{11}, i_{11}, j_{11}, k_{11}, l_{11}, m_{11}, n_{11}, p_{11}, q_{11}, f_1, r_{11}, s_{11}, t_{11}, a_{12}, b_{12}, c_{12}, d_{12}, e_{12}, f_{12}, g_{12}, g_1, h_{12}, i_{12}, j_{12}, k_{12}, l_{12}, m_{12}, n_{12}, p_{12}, q_{12}, r_{12}, h_1, s_{12}, t_{12}, a_{13}, b_{13}, c_{13}, d_{13}, e_{13}, f_{13}, g_{13}, h_{13}, i_1, i_{13}, j_{13}, k_{13}, l_{13}, m_{13}, n_{13}, p_{13}, q_{13}, r_{13}, s_{13}, j_1, t_{13}, a_{14}, b_{14}, c_{14}, d_{14}, e_{14}, f_{14}, g_{14}, h_{14}, i_{14}, k_1, j_{14}, k_{14}, l_{14}, m_{14}, n_{14}, p_{14}, q_{14}, r_{14}, s_{14}, t_{14}, l_1, a_{15}, b_{15}, c_{15}, d_{15}, e_{15}, f_{15}, g_{15}, h_{15}, i_{15}, j_{15}, m_1, k_{15}, l_{15}, m_{15}, n_{15}, p_{15}, q_{15}, r_{15}, s_{15}, t_{15}, a_{16}, e, n_1, b_{16}, C, D, E, F, G, H, I, J, K, q_1, L, M, N, O, P, Q, R, S, T, U, r_1, V, W, Z, A_1, B_1, C_1, D_1, E_1, F_1, H_1, a_2, I_1, J_1, K_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1, b_2, S_1, T_1, U_1, V_1, W_1, Z_1, A_2, B_2, C_2, D_2, c_2, E_2, F_2, G_2, H_2, I_2, J_2, K_2, L_2, M_2, N_2, d_2, O_2, P_2, Q_2, R_2, S_2, T_2, U_2, V_2, W_2, Z_2, e_2, A_3, B_3, C_3, D_3, E_3, F_3, G_3, H_3, I_3, J_3, f_2, K_3, L_3, M_3, N_3, O_3, P_3, Q_3, R_3, S_3, T_3, g_2, U_3, V_3, W_3, Z_3, A_4, B_4, C_4, D_4, E_4, F_4, f, h_2, G_4, H_4, I_4, J_4, K_4, L_4, M_4, N_4, O_4, P_4, i_2, Q_4, R_4, S_4, T_4, U_4, V_4, W_4, Z_4, A_5, B_5, j_2, C_5, D_5, E_5, F_5, G_5, H_5, I_5, J_5, K_5, L_5, k_2, M_5, N_5, O_5, P_5, Q_5, R_5, S_5, T_5, U_5, V_5, l_2, W_5, Z_5, A_6, B_6, C_6, D_6, E_6, F_6, G_6, H_6, m_2, I_6, J_6, K_6, L_6, M_6, N_6, O_6, P_6, Q_6, R_6, n_2, S_6, T_6, U_6, V_6, W_6, Z_6, A_7, B_7, C_7, D_7, p_2, E_7, F_7, G_7, H_7, I_7, J_7, K_7, L_7, M_7, N_7, q_2, O_7, P_7, Q_7, R_7, S_7, T_7, U_7, V_7, W_7, Z_7, r_2, A_8, B_8, C_8, D_8, E_8, F_8, G_8, H_8, I_8, J_8, g, s_2, K_8, L_8, M_8, N_8, O_8, P_8, Q_8, R_8, S_8, T_8, t_2, U_8, V_8, W_8, Z_8, A_9, B_9, C_9, D_9, E_9, F_9, a_3, G_9, H_9, I_9, J_9, K_9, L_9, M_9, N_9, O_9, P_9, b_3, Q_9, R_9, S_9, T_9, U_9, V_9, W_9, Z_9, A_{10}, B_{10}, c_3, C_{10}, D_{10}, E_{10}, F_{10}, G_{10}, H_{10}, I_{10}, J_{10}, K_{10}, L_{10}, d_3, M_{10}, N_{10}, O_{10}, P_{10}, Q_{10}, R_{10}, S_{10}, T_{10}, U_{10}, V_{10}, e_3, W_{10}, Z_{10}, A_{11}, B_{11}, C_{11}, D_{11}, E_{11}, F_{11}, G_{11}, H_{11}, f_3, I_{11}, J_{11}, K_{11}, L_{11}, M_{11}, N_{11}, O_{11}, P_{11}, Q_{11}, R_{11}, g_3, S_{11}, T_{11}, U_{11}, V_{11}, W_{11}, Z_{11}, A_{12}, B_{12}, C_{12}, D_{12}, h_3, E_{12}, F_{12}, G_{12}, H_{12}, I_{12}, J_{12}, K_{12}, L_{12}, M_{12}, h, i_3, j_3, k_3, l_3, m_3, n_3, p_3, q_3, r_3, s_3, i, t_3, a_4, b_4, c_4, d_4, e_4, f_4, g_4, h_4, i_4, j, j_4, k_4, l_4, m_4, n_4, p_4, q_4, r_4, s_4, t_4, k, a_5, b_5, c_5, d_5, e_5, f_5, g_5, h_5, i_5, j_5
  5. Area(c) → o
  6. OrthogonalLine(A, b) → p_1
  7. Intersect(p_1, b) → G_1
  8. Segment(A, G_1) → Apothem
  9. Segment(A, B) → RadiusOfCircle
  10. Angle(G_1, B, A) → β
  11. Segment(A, B') → s_1
  12. Segment(G_1, B) → t_1
  13. Segment(A, C) → d_{16}
  14. Angle(B', A, C) → θ
  15. Circle(B, 3) → e_{16}
  16. Intersect(e_{16}, RadiusOfCircle, 1) → O_{12}
  17. Ray(B, B') → c_{16}
  18. Point(d) → N_{12}
  19. Point(c_{16}) → P_{12}
  20. Angle(P_{12}, B', N_{12}) → γ
  21. Angle(C, B', B) → δ
  22. Ray(B, B') → f_{16}
  23. LaTeX(δ, true, true) → text6
  24. LaTeX(θ, true, true) → text7
  25. LaTeX(γ, true, true) → text10
  26. LaTeX(β, true, true) → text11