Deriving the Sum Formula for First n Natural Numbers

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2014-03-27 04:28

This example demonstrates the geometric and algebraic derivation of the sum formula for the first n natural numbers through dynamic constructions. Key math concepts include sequence summation, algebraic manipulation, and geometric visualization. GeoGebra commands used are Line, Circle, Intersect, Polygon, Segment, Vector, Translate, Point, and OrthogonalLine. Suitable for high school mathematics education to understand the geometric interpretation and algebraic proof of sum formulas.

GeoGebra Commands & Parameters

Line(A, xAxis) → b
Line(A, yAxis) → c
Circle(A, a) → d
Intersect(d, b, 2) → C
Line(C, c) → e
Intersect(d, c, 1) → D
Line(D, b) → f
Circle(C, A) → g
Circle(D, A) → h
Intersect(g, b, 2) → E
Circle(E, C) → k
Intersect(h, c, 2) → H
Circle(H, D) → r
Line(E, e) → i
Line(H, f) → j
Intersect(r, c, 2) → I
Line(I, j) → l
Intersect(k, b, 2) → G
Circle(G, E) → p
Intersect(p, b, 2) → J
Circle(J, G) → q
Intersect(q, b, 2) → K
Circle(K, J) → s
Intersect(s, b, 2) → L
Circle(L, K) → t
Intersect(t, b, 2) → M
Circle(M, L) → c_1
Intersect(c_1, b, 2) → N
Circle(N, M) → d_1
Intersect(d_1, b, 2) → O
Circle(O, N) → e_1
Intersect(e_1, b, 2) → P
Circle(P, O) → f_1
Circle(I, H) → g_1
Intersect(g_1, c, 2) → Q
Circle(Q, I) → h_1
Intersect(h_1, c, 2) → R
Circle(R, Q) → k_1
Intersect(k_1, c, 2) → S
Circle(S, R) → p_1
Intersect(p_1, c, 2) → T
Circle(T, S) → q_1
Intersect(q_1, c, 2) → U
Circle(U, T) → r_1
Line(Q, f) → m
Line(R, l) → n
Line(S, m) → a_1
Line(T, n) → b_1
Line(U, a_1) → i_1
Line(G, i) → j_1
Line(J, i) → l_1
Line(K, j_1) → m_1
Line(L, m_1) → n_1
Line(M, n_1) → s_1
Line(N, s_1) → t_1
Line(O, t_1) → a_2
Line(P, a_2) → b_2
Circle(P, O) → c_2
Intersect(c_2, b, 2) → B
Circle(B, P) → d_2
Intersect(d_2, b, 2) → F
Circle(F, B) → e_2
Intersect(e_2, b, 2) → V
Circle(V, F) → f_2
Intersect(f_2, b, 2) → W
Circle(W, V) → g_2
Intersect(g_2, b, 2) → Z
Circle(Z, W) → h_2
Circle(U, T) → c_3
Intersect(c_3, c, 2) → G_1
Circle(G_1, U) → d_3
Intersect(d_3, c, 2) → H_1
Circle(H_1, G_1) → e_3
Intersect(e_3, c, 2) → I_1
Circle(I_1, H_1) → f_3
Intersect(f_3, c, 2) → J_1
Circle(J_1, I_1) → g_3
Intersect(g_3, c, 2) → K_1
Circle(K_1, J_1) → h_3
Intersect(h_3, c, 2) → L_1
Circle(L_1, K_1) → k_3
Intersect(k_3, c, 2) → M_1
Line(G_1, i_1) → i_2
Line(H_1, i_2) → j_2
Line(I_1, j_2) → l_2
Line(J_1, l_2) → m_2
Line(K_1, m_2) → n_2
Line(L_1, n_2) → a_3
Line(M_1, a_3) → b_3
Line(B, b_2) → i_3
Line(F, i_3) → j_3
Line(V, j_3) → l_3
Line(W, j_3) → m_3
Line(Z, l_3) → n_3
Intersect(e, f) → O_1
Polygon(A, D, O_1, C) → mnohoúhelník1, a_5, d_4, o_1, c_4
Intersect(f, i) → P_1
Intersect(f, j_1) → Q_1
Polygon(E, P_1, Q_1, G) → mnohoúhelník2, e_4, p_4, q_4, g_4
Intersect(j, j_1) → R_1
Intersect(i, j) → S_1
Polygon(P_1, Q_1, R_1, S_1) → mnohoúhelník3, p_5, q_5, r_4, s_4
Intersect(f, m_1) → T_1
Intersect(f, l_1) → U_1
Polygon(J, K, T_1, U_1) → mnohoúhelník4, j_4, k_4, t_4, u_1
Intersect(j, l_1) → V_1
Intersect(j, m_1) → W_1
Polygon(U_1, V_1, W_1, T_1) → mnohoúhelník5, u_2, v_1, w_1, t_5
Intersect(l, m_1) → Z_1
Intersect(l, l_1) → A_2
Polygon(V_1, W_1, Z_1, A_2) → mnohoúhelník6, v_2, w_2, z_1, a_6
Intersect(f, s_1) → B_2
Intersect(f, n_1) → C_2
Polygon(L, M, B_2, C_2) → mnohoúhelník7, l_4, m_4, b_5, c_5
Intersect(j, s_1) → D_2
Intersect(j, n_1) → E_2
Polygon(C_2, B_2, D_2, E_2) → mnohoúhelník8, c_6, b_6, d_5, e_5
Intersect(l, s_1) → F_2
Intersect(l, n_1) → G_2
Polygon(E_2, D_2, F_2, G_2) → mnohoúhelník9, e_6, d_6, f_4, g_5
Intersect(m, n_1) → H_2
Intersect(m, s_1) → I_2
Polygon(F_2, G_2, H_2, I_2) → mnohoúhelník10, f_5, g_6, h_4, i_4
Intersect(f, a_2) → J_2
Intersect(f, t_1) → K_2
Polygon(N, O, J_2, K_2) → mnohoúhelník11, n_4, o, j_5, k_5
Intersect(j, a_2) → L_2
Intersect(j, t_1) → M_2
Polygon(K_2, J_2, L_2, M_2) → mnohoúhelník12, k_6, j_6, l_5, m_5
Intersect(l, a_2) → N_2
Intersect(l, t_1) → O_2
Polygon(M_2, L_2, N_2, O_2) → mnohoúhelník13, m_6, l_6, n_5, o_2
Intersect(m, a_2) → P_2
Intersect(m, t_1) → Q_2
Polygon(O_2, N_2, P_2, Q_2) → mnohoúhelník14, o_3, n_6, p_6, q_6
Intersect(n, a_2) → R_2
Intersect(n, t_1) → S_2
Polygon(Q_2, P_2, R_2, S_2) → mnohoúhelník15, q_7, p_7, r_5, s_5
Intersect(f, i_3) → T_2
Intersect(f, b_2) → U_2
Polygon(P, B, T_2, U_2) → mnohoúhelník16, p_8, b_7, t_6, u_3
Intersect(j, i_3) → V_2
Intersect(j, b_2) → W_2
Polygon(U_2, T_2, V_2, W_2) → mnohoúhelník17, u_4, t_7, v_3, w_3
Intersect(l, i_3) → Z_2
Intersect(l, b_2) → A_3
Polygon(W_2, V_2, Z_2, A_3) → mnohoúhelník18, w_4, v_4, z_2, a_7
Intersect(m, i_3) → B_3
Intersect(m, b_2) → C_3
Polygon(A_3, Z_2, B_3, C_3) → mnohoúhelník19, a_8, z_3, b_8, c_7
Intersect(n, i_3) → D_3
Intersect(n, b_2) → E_3
Polygon(C_3, B_3, D_3, E_3) → mnohoúhelník20, c_8, b_9, d_7, e_7
Intersect(a_1, i_3) → F_3
Intersect(a_1, b_2) → G_3
Polygon(E_3, D_3, F_3, G_3) → mnohoúhelník21, e_8, d_8, f_6, g_7
Intersect(f, l_3) → H_3
Intersect(f, j_3) → I_3
Polygon(F, V, H_3, I_3) → mnohoúhelník22, f_7, v, h_5, i_5
Intersect(j, l_3) → J_3
Intersect(j, j_3) → K_3
Polygon(I_3, H_3, J_3, K_3) → mnohoúhelník23, i_6, h_6, j_7, k_7
Intersect(l, l_3) → L_3
Intersect(l, j_3) → M_3
Polygon(K_3, J_3, L_3, M_3) → mnohoúhelník24, k_8, j_8, l_7, m_7
Intersect(m, l_3) → N_3
Intersect(m, j_3) → O_3
Polygon(M_3, L_3, N_3, O_3) → mnohoúhelník25, m_8, l_8, n_7, o_4
Intersect(n, l_3) → P_3
Intersect(n, j_3) → Q_3
Polygon(O_3, N_3, P_3, Q_3) → mnohoúhelník26, o_5, n_8, p_9, q_8
Intersect(a_1, l_3) → R_3
Intersect(a_1, j_3) → S_3
Polygon(Q_3, P_3, R_3, S_3) → mnohoúhelník27, q_9, p_{10}, r_6, s_6
Intersect(b_1, l_3) → T_3
Intersect(b_1, j_3) → U_3
Polygon(S_3, R_3, T_3, U_3) → mnohoúhelník28, s_7, r_7, t_8, u_5
Circle(M_1, L_1) → h_7
Intersect(h_7, c, 2) → V_3
Circle(V_3, M_1) → f_8
Line(V_3, b_3) → i_7
Intersect(f_8, c, 2) → W_3
Line(W_3, i_7) → g_8
Intersect(f, n_3) → Z_3
Intersect(f, m_3) → A_4
Polygon(W, Z, Z_3, A_4) → mnohoúhelník29, w_{13}, z, z_4, a_9
Intersect(j, n_3) → B_4
Intersect(j, m_3) → C_4
Polygon(A_4, Z_3, B_4, C_4) → mnohoúhelník30, a_{10}, z_5, b_{10}, c_9
Intersect(l, n_3) → D_4
Intersect(l, m_3) → E_4
Polygon(C_4, B_4, D_4, E_4) → mnohoúhelník31, c_{10}, b_{11}, d_9, e_9
Intersect(m, n_3) → F_4
Intersect(m, m_3) → G_4
Polygon(E_4, D_4, F_4, G_4) → mnohoúhelník32, e_{10}, d_{10}, f_9, g_9
Intersect(n, n_3) → H_4
Intersect(n, m_3) → I_4
Polygon(G_4, F_4, H_4, I_4) → mnohoúhelník33, g_{10}, f_{10}, h_8, i_8
Intersect(a_1, n_3) → J_4
Intersect(a_1, m_3) → K_4
Polygon(I_4, H_4, J_4, K_4) → mnohoúhelník34, i_9, h_9, j_9, k_9
Intersect(b_1, n_3) → L_4
Intersect(b_1, m_3) → M_4
Polygon(K_4, J_4, L_4, M_4) → mnohoúhelník35, k_{10}, j_{10}, l_9, m_9
Intersect(i_1, n_3) → N_4
Intersect(i_1, m_3) → O_4
Polygon(M_4, L_4, N_4, O_4) → mnohoúhelník36, m_{10}, l_{10}, n_9, o_6
Intersect(e, g_8) → P_4
Intersect(e, i_7) → Q_4
Polygon(W_3, P_4, Q_4, V_3) → mnohoúhelník37, w_5, p_{11}, q_{10}, v_5
Intersect(i, g_8) → R_4
Intersect(j_1, g_8) → S_4
Intersect(j_1, i_7) → T_4
Intersect(i, i_7) → U_4
Polygon(R_4, S_4, T_4, U_4) → mnohoúhelník38, r_8, s_8, t_9, u_6
Intersect(j_1, b_3) → V_4
Intersect(i, b_3) → W_4
Polygon(U_4, T_4, V_4, W_4) → mnohoúhelník39, u_7, t_{10}, v_6, w_6
Intersect(l_1, g_8) → Z_4
Intersect(m_1, g_8) → A_5
Intersect(m_1, i_7) → B_5
Intersect(l_1, i_7) → C_5
Polygon(Z_4, A_5, B_5, C_5) → mnohoúhelník40, z_6, a_{11}, b_{12}, c_{11}
Intersect(m_1, b_3) → D_5
Intersect(l_1, b_3) → E_5
Polygon(C_5, B_5, D_5, E_5) → mnohoúhelník41, c_{12}, b_{13}, d_{11}, e_{11}
Intersect(m_1, a_3) → F_5
Intersect(l_1, a_3) → G_5
Polygon(E_5, D_5, F_5, G_5) → mnohoúhelník42, e_{12}, d_{12}, f_{11}, g_{11}
Intersect(m_1, n_2) → H_5
Intersect(l_1, n_2) → I_5
Intersect(n_1, g_8) → J_5
Intersect(s_1, g_8) → K_5
Intersect(s_1, i_7) → L_5
Intersect(n_1, i_7) → M_5
Polygon(J_5, K_5, L_5, M_5) → mnohoúhelník43, j_{11}, k_{11}, l_{11}, m_{11}
Intersect(s_1, b_3) → N_5
Intersect(n_1, b_3) → O_5
Polygon(M_5, L_5, N_5, O_5) → mnohoúhelník44, m_{12}, l_{12}, n_{10}, o_7
Intersect(s_1, a_3) → P_5
Intersect(n_1, a_3) → Q_5
Polygon(O_5, N_5, P_5, Q_5) → mnohoúhelník45, o_8, n_{11}, p_{12}, q_{11}
Intersect(s_1, n_2) → R_5
Intersect(n_1, n_2) → S_5
Polygon(Q_5, P_5, R_5, S_5) → mnohoúhelník46, q_{12}, p_{13}, r_9, s_9
Intersect(t_1, g_8) → T_5
Intersect(a_2, g_8) → U_5
Intersect(a_2, i_7) → V_5
Intersect(t_1, i_7) → W_5
Polygon(T_5, U_5, V_5, W_5) → mnohoúhelník47, t_{11}, u_8, v_7, w_7
Intersect(a_2, b_3) → Z_5
Intersect(t_1, b_3) → A_6
Polygon(W_5, V_5, Z_5, A_6) → mnohoúhelník48, w_8, v_8, z_7, a_{12}
Intersect(a_2, a_3) → B_6
Intersect(t_1, a_3) → C_6
Polygon(A_6, Z_5, B_6, C_6) → mnohoúhelník49, a_{13}, z_8, b_{14}, c_{13}
Intersect(a_2, n_2) → D_6
Intersect(t_1, n_2) → E_6
Polygon(C_6, B_6, D_6, E_6) → mnohoúhelník50, c_{14}, b_{15}, d_{13}, e_{13}
Intersect(e, b_3) → H_6
Polygon(V_3, Q_4, H_6, M_1) → mnohoúhelník52, v_9, q_{13}, h_{10}, m_{13}
Intersect(e, a_3) → I_6
Polygon(M_1, H_6, I_6, L_1) → mnohoúhelník53, m_{14}, h_{11}, i_{10}, l_{13}
Intersect(e, n_2) → J_6
Polygon(L_1, I_6, J_6, K_1) → mnohoúhelník54, l_{14}, i_{11}, j_{12}, k_{12}
Intersect(e, m_2) → K_6
Polygon(K_1, J_6, K_6, J_1) → mnohoúhelník55, k_{13}, j_{13}, k_{14}, j_{14}
Intersect(e, l_2) → L_6
Polygon(J_1, K_6, L_6, I_1) → mnohoúhelník56, j_{15}, k_{15}, l_{15}, i_{12}
Intersect(e, j_2) → M_6
Polygon(I_1, L_6, M_6, H_1) → mnohoúhelník57, i_{13}, l_{16}, m_{15}, h_{12}
Intersect(e, i_2) → N_6
Polygon(H_1, M_6, N_6, G_1) → mnohoúhelník58, h_{13}, m_{16}, n_{12}, g_{13}
Intersect(j_1, a_3) → O_6
Intersect(i, a_3) → P_6
Polygon(W_4, V_4, O_6, P_6) → mnohoúhelník59, w_9, v_{10}, o_9, p_{14}
Intersect(j_1, n_2) → Q_6
Intersect(i, n_2) → R_6
Polygon(P_6, O_6, Q_6, R_6) → mnohoúhelník60, p_{15}, o_{10}, q_{14}, r_{10}
Intersect(j_1, m_2) → S_6
Intersect(i, m_2) → T_6
Polygon(R_6, Q_6, S_6, T_6) → mnohoúhelník61, r_{11}, q_{15}, s_{10}, t_{12}
Intersect(j_1, l_2) → U_6
Intersect(i, l_2) → V_6
Polygon(T_6, S_6, U_6, V_6) → mnohoúhelník62, t_{13}, s_{11}, u_9, v_{11}
Intersect(j_1, j_2) → W_6
Intersect(i, j_2) → Z_6
Polygon(V_6, U_6, W_6, Z_6) → mnohoúhelník63, v_{12}, u_{10}, w_{10}, z_9
Intersect(m_1, n_2) → A_7
Intersect(l_1, n_2) → B_7
Polygon(G_5, F_5, A_7, B_7) → mnohoúhelník64, g_{14}, f_{13}, a_{14}, b_{16}
Intersect(m_1, m_2) → C_7
Intersect(l_1, m_2) → D_7
Polygon(B_7, A_7, C_7, D_7) → mnohoúhelník65, b_{17}, a_{15}, c_{15}, d_{15}
Intersect(m_1, l_2) → E_7
Intersect(l_1, l_2) → F_7
Polygon(D_7, C_7, E_7, F_7) → mnohoúhelník66, d_{16}, c_{16}, e_{15}, f_{14}
Intersect(s_1, m_2) → G_7
Intersect(n_1, m_2) → H_7
Polygon(S_5, R_5, G_7, H_7) → mnohoúhelník67, s_{12}, r_{12}, g_{15}, h_{14}
Intersect(b_2, g_8) → F_6
Intersect(i_3, g_8) → G_6
Intersect(i_3, i_7) → I_7
Intersect(b_2, i_7) → J_7
Polygon(F_6, G_6, I_7, J_7) → mnohoúhelník51, f_{12}, g_{12}, i_{14}, j_{16}
Intersect(b_3, i_3) → K_7
Intersect(b_2, b_3) → L_7
Polygon(J_7, I_7, K_7, L_7) → mnohoúhelník68, j_{17}, i_{15}, k_{16}, l_{17}
Intersect(a_3, i_3) → M_7
Intersect(b_2, a_3) → N_7
Polygon(L_7, K_7, M_7, N_7) → mnohoúhelník69, l_{18}, k_{17}, m_{17}, n_{13}
Intersect(j_3, g_8) → O_7
Intersect(l_3, g_8) → P_7
Intersect(l_3, i_7) → Q_7
Intersect(j_3, i_7) → R_7
Polygon(O_7, P_7, Q_7, R_7) → mnohoúhelník70, o_{11}, p_{16}, q_{16}, r_{13}
Intersect(b_3, l_3) → S_7
Intersect(b_3, j_3) → T_7
Polygon(R_7, Q_7, S_7, T_7) → mnohoúhelník71, r_{14}, q_{17}, s_{13}, t_{14}
Intersect(m_3, g_8) → U_7
Intersect(n_3, g_8) → V_7
Intersect(n_3, i_7) → W_7
Intersect(m_3, i_7) → Z_7
Polygon(U_7, V_7, W_7, Z_7) → mnohoúhelník72, u_{11}, v_{13}, w_{11}, z_{10}
Circle(A, 0.4) → d_{14}
Circle(Z, 0.4) → k_2
Circle(W_3, 0.4) → p_2
Circle(V_7, 0.4) → q_2
Intersect(d_{14}, b, 1) → A_1
Intersect(d_{14}, c, 2) → B_1
Intersect(k_2, n_3, 2) → C_1
Intersect(k_2, b, 2) → D_1
Intersect(p_2, c, 1) → E_1
Line(A_1, e) → r_2
Line(E_1, m_2) → s_2
Line(D_1, n_3) → t_2
Line(B_1, b) → p_3
Intersect(r_2, p_3) → F_1
Intersect(t_2, p_3) → N_1
Segment(F_1, N_1) → q_3
Intersect(r_2, s_2) → A_8
Segment(F_1, A_8) → r_3
Intersect(s_2, t_2) → B_8
Segment(A_8, B_8) → s_3
Segment(B_8, N_1) → t_3
Intersect(e, p_3) → C_8
Intersect(i, p_3) → D_8
Intersect(j_1, p_3) → E_8
Intersect(l_1, p_3) → F_8
Intersect(m_1, p_3) → G_8
Intersect(n_1, p_3) → H_8
Intersect(s_1, p_3) → I_8
Intersect(t_1, p_3) → J_8
Intersect(a_2, p_3) → K_8
Intersect(b_2, p_3) → L_8
Intersect(i_3, p_3) → M_8
Intersect(j_3, p_3) → N_8
Intersect(l_3, p_3) → O_8
Intersect(m_3, p_3) → P_8
Intersect(f, r_2) → Q_8
Intersect(j, r_2) → R_8
Intersect(l, r_2) → S_8
Intersect(m, r_2) → T_8
Intersect(n, r_2) → U_8
Intersect(a_1, r_2) → V_8
Intersect(b_1, r_2) → W_8
Intersect(i_1, r_2) → Z_8
Intersect(i_2, r_2) → A_9
Intersect(j_2, r_2) → B_9
Intersect(l_2, r_2) → C_9
Intersect(m_2, r_2) → D_9
Intersect(n_2, r_2) → E_9
Intersect(a_3, r_2) → F_9
Intersect(b_3, r_2) → G_9
Intersect(i_7, r_2) → H_9
Intersect(g_8, r_2) → I_9
Intersect(e, s_2) → J_9
Intersect(i, s_2) → K_9
Intersect(j_1, s_2) → L_9
Intersect(l_1, s_2) → M_9
Intersect(m_1, s_2) → N_9
Intersect(n_1, s_2) → O_9
Intersect(s_1, s_2) → P_9
Intersect(t_1, s_2) → Q_9
Intersect(a_2, s_2) → R_9
Intersect(b_2, s_2) → S_9
Intersect(i_3, s_2) → T_9
Intersect(j_3, s_2) → U_9
Intersect(l_3, s_2) → V_9
Intersect(m_3, s_2) → W_9
Intersect(n_3, s_2) → Z_9
Intersect(g_8, t_2) → A_{10}
Intersect(i_7, t_2) → B_{10}
Intersect(b_3, t_2) → C_{10}
Intersect(a_3, t_2) → D_{10}
Intersect(n_2, t_2) → E_{10}
Intersect(m_2, t_2) → F_{10}
Intersect(l_2, t_2) → G_{10}
Intersect(j_2, t_2) → H_{10}
Intersect(i_2, t_2) → I_{10}
Intersect(i_1, t_2) → J_{10}
Intersect(b_1, t_2) → K_{10}
Intersect(a_1, t_2) → L_{10}
Intersect(n, t_2) → M_{10}
Intersect(m, t_2) → N_{10}
Intersect(l, t_2) → O_{10}
Intersect(j, t_2) → P_{10}
Intersect(f, t_2) → Q_{10}
Vector(G_1, D) → u
Translate(H_1, u_x) → R_{10}
Translate(M_6, u_x) → S_{10}
Translate(N_6, u_x) → T_{10}
Translate(G_1, u_x) → U_{10}
Polygon(R_{10}, S_{10}, T_{10}, U_{10}) → mnohoúhelník58', r_{15}, s_{14}, t_{15}, u_{14}
Translate(I_1, u_x) → V_{10}
Translate(L_6, u_x) → W_{10}
Translate(M_6, u_x) → Z_{10}
Translate(H_1, u_x) → A_{11}
Polygon(V_{10}, W_{10}, Z_{10}, A_{11}) → mnohoúhelník57', v_{14}, w_{14}, z_{13}, a_{16}
Translate(J_1, u_x) → B_{11}
Translate(K_6, u_x) → C_{11}
Translate(L_6, u_x) → D_{11}
Translate(I_1, u_x) → E_{11}
Intersect(g_{13}, n_{12}) → F_{11}
Intersect(n_{12}, m_{16}) → G_{11}
Segment(F_{11}, G_{11}) → e_{14}
Intersect(p_{11}, w_5) → H_{11}
Segment(G_{11}, H_{11}) → n_{14}
Intersect(v_5, w_5) → I_{11}
Segment(H_{11}, I_{11}) → f_{15}
Segment(I_{11}, F_{11}) → h_{15}
Translate(J_1, u_x) → J_{11}
Translate(K_6, u_x) → K_{11}
Translate(L_6, u_x) → L_{11}
Translate(I_1, u_x) → M_{11}
Polygon(J_{11}, K_{11}, L_{11}, M_{11}) → mnohoúhelník56', j_{18}, k_{18}, l_{19}, m_{18}
Translate(K_1, u_x) → N_{11}
Translate(J_6, u_x) → O_{11}
Translate(K_6, u_x) → P_{11}
Translate(J_1, u_x) → Q_{11}
Polygon(N_{11}, O_{11}, P_{11}, Q_{11}) → mnohoúhelník55', n_{15}, o_{12}, p_{17}, q_{18}
Translate(L_1, u_x) → R_{11}
Translate(I_6, u_x) → S_{11}
Translate(J_6, u_x) → T_{11}
Translate(K_1, u_x) → U_{11}
Polygon(R_{11}, S_{11}, T_{11}, U_{11}) → mnohoúhelník54', r_{16}, s_{15}, t_{16}, u_{15}
Translate(M_1, u_x) → V_{11}
Translate(H_6, u_x) → W_{11}
Translate(I_6, u_x) → Z_{11}
Translate(L_1, u_x) → A_{12}
Polygon(V_{11}, W_{11}, Z_{11}, A_{12}) → mnohoúhelník53', v_{15}, w_{15}, z_{14}, a_{17}
Translate(V_3, u_x) → B_{12}
Translate(Q_4, u_x) → C_{12}
Translate(H_6, u_x) → D_{12}
Translate(M_1, u_x) → E_{12}
Polygon(B_{12}, C_{12}, D_{12}, E_{12}) → mnohoúhelník52', b_{18}, c_{17}, d_{17}, e_{16}
Translate(W_3, u_x) → F_{12}
Translate(P_4, u_x) → G_{12}
Translate(Q_4, u_x) → H_{12}
Translate(V_3, u_x) → I_{12}
Polygon(F_{12}, G_{12}, H_{12}, I_{12}) → mnohoúhelník37', f_{16}, g_{16}, h_{16}, i_{16}
Translate(R_{10}, u_x) → J_{12}
Translate(S_{10}, u_x) → K_{12}
Translate(T_{10}, u_x) → L_{12}
Translate(U_{10}, u_x) → M_{12}
Polygon(J_{12}, K_{12}, L_{12}, M_{12}) → mnohoúhelník58'', j_{19}, k_{19}, l_{20}, m_{19}
Translate(V_{10}, u_x) → N_{12}
Translate(W_{10}, u_x) → O_{12}
Translate(Z_{10}, u_x) → P_{12}
Translate(A_{11}, u_x) → Q_{12}
Polygon(N_{12}, O_{12}, P_{12}, Q_{12}) → mnohoúhelník57'', n_{16}, o_{13}, p_{18}, q_{19}
Translate(J_{11}, u_x) → R_{12}
Translate(K_{11}, u_x) → S_{12}
Translate(L_{11}, u_x) → T_{12}
Translate(M_{11}, u_x) → U_{12}
Polygon(R_{12}, S_{12}, T_{12}, U_{12}) → mnohoúhelník56'', r_{17}, s_{16}, t_{17}, u_{16}
Translate(N_{11}, u_x) → V_{12}
Translate(O_{11}, u_x) → W_{12}
Translate(P_{11}, u_x) → Z_{12}
Translate(Q_{11}, u_x) → A_{13}
Polygon(V_{12}, W_{12}, Z_{12}, A_{13}) → mnohoúhelník55'', v_{16}, w_{16}, z_{15}, a_{18}
Translate(R_{11}, u_x) → B_{13}
Translate(S_{11}, u_x) → C_{13}
Translate(T_{11}, u_x) → D_{13}
Translate(U_{11}, u_x) → E_{13}
Polygon(B_{13}, C_{13}, D_{13}, E_{13}) → mnohoúhelník54'', b_{19}, c_{18}, d_{18}, e_{17}
Translate(V_{11}, u_x) → F_{13}
Translate(W_{11}, u_x) → G_{13}
Translate(Z_{11}, u_x) → H_{13}
Translate(A_{12}, u_x) → I_{13}
Polygon(F_{13}, G_{13}, H_{13}, I_{13}) → mnohoúhelník53'', f_{17}, g_{17}, h_{17}, i_{17}
Translate(B_{12}, u_x) → J_{13}
Translate(C_{12}, u_x) → K_{13}
Translate(D_{12}, u_x) → L_{13}
Translate(E_{12}, u_x) → M_{13}
Polygon(J_{13}, K_{13}, L_{13}, M_{13}) → mnohoúhelník52'', j_{20}, k_{20}, l_{21}, m_{20}
Translate(F_{12}, u_x) → N_{13}
Translate(G_{12}, u_x) → O_{13}
Translate(H_{12}, u_x) → P_{13}
Translate(I_{12}, u_x) → Q_{13}
Polygon(N_{13}, O_{13}, P_{13}, Q_{13}) → mnohoúhelník37'', n_{17}, o_{14}, p_{19}, q_{20}
Translate(F_{11}, u_x) → R_{13}
Translate(G_{11}, u_x) → S_{13}
Segment(R_{13}, S_{13}) → s_{17}
Translate(J_{12}, u_x) → T_{13}
Translate(K_{12}, u_x) → U_{13}
Translate(L_{12}, u_x) → V_{13}
Translate(M_{12}, u_x) → W_{13}
Polygon(T_{13}, U_{13}, V_{13}, W_{13}) → mnohoúhelník58''', t_{18}, u_{17}, v_{17}, w_{17}
Translate(N_{12}, u_x) → Z_{13}
Translate(O_{12}, u_x) → A_{14}
Translate(P_{12}, u_x) → B_{14}
Translate(Q_{12}, u_x) → C_{14}
Polygon(Z_{13}, A_{14}, B_{14}, C_{14}) → mnohoúhelník57''', z_{16}, a_{19}, b_{20}, c_{19}
Translate(R_{12}, u_x) → D_{14}
Translate(S_{12}, u_x) → E_{14}
Translate(T_{12}, u_x) → F_{14}
Translate(U_{12}, u_x) → G_{14}
Polygon(D_{14}, E_{14}, F_{14}, G_{14}) → mnohoúhelník56''', d_{19}, e_{18}, f_{18}, g_{18}
Translate(V_{12}, u_x) → H_{14}
Translate(W_{12}, u_x) → I_{14}
Translate(Z_{12}, u_x) → J_{14}
Translate(A_{13}, u_x) → K_{14}
Polygon(H_{14}, I_{14}, J_{14}, K_{14}) → mnohoúhelník55''', h_{18}, i_{18}, j_{21}, k_{21}
Translate(B_{13}, u_x) → L_{14}
Translate(C_{13}, u_x) → M_{14}
Translate(D_{13}, u_x) → N_{14}
Translate(E_{13}, u_x) → O_{14}
Polygon(L_{14}, M_{14}, N_{14}, O_{14}) → mnohoúhelník54''', l_{22}, m_{21}, n_{18}, o_{15}
Translate(F_{13}, u_x) → P_{14}
Translate(G_{13}, u_x) → Q_{14}
Translate(H_{13}, u_x) → R_{14}
Translate(I_{13}, u_x) → S_{14}
Polygon(P_{14}, Q_{14}, R_{14}, S_{14}) → mnohoúhelník53''', p_{20}, q_{21}, r_{18}, s_{18}
Translate(J_{13}, u_x) → T_{14}
Translate(K_{13}, u_x) → U_{14}
Translate(L_{13}, u_x) → V_{14}
Translate(M_{13}, u_x) → W_{14}
Polygon(T_{14}, U_{14}, V_{14}, W_{14}) → mnohoúhelník52''', t_{19}, u_{18}, v_{18}, w_{18}
Translate(N_{13}, u_x) → Z_{14}
Translate(O_{13}, u_x) → A_{15}
Translate(P_{13}, u_x) → B_{15}
Translate(Q_{13}, u_x) → C_{15}
Polygon(Z_{14}, A_{15}, B_{15}, C_{15}) → mnohoúhelník37''', z_{17}, a_{20}, b_{21}, c_{20}
Translate(F_{11}, u_x) → D_{15}
Translate(G_{11}, u_x) → E_{15}
Segment(D_{15}, E_{15}) → e_{19}
Point(a_1) → F_{15}
OrthogonalLine(F_{15}, a_1) → f_{19}
Circle(F_{15}, a) → g_{19}
Intersect(g_{19}, a_1, 2) → G_{15}
OrthogonalLine(G_{15}, a_1) → h_{19}
Intersect(b_1, f_{19}) → H_{15}
Intersect(b_1, h_{19}) → I_{15}
Translate(R_4, u_x) → J_{15}
Translate(S_4, u_x) → K_{15}
Translate(T_4, u_x) → L_{15}
Translate(U_4, u_x) → M_{15}
Polygon(J_{15}, K_{15}, L_{15}, M_{15}) → mnohoúhelník38', j_{22}, k_{22}, l_{23}, m_{22}
Translate(U_4, u_x) → N_{15}
Translate(T_4, u_x) → O_{15}
Translate(V_4, u_x) → P_{15}
Translate(W_4, u_x) → Q_{15}
Polygon(N_{15}, O_{15}, P_{15}, Q_{15}) → mnohoúhelník39', n_{19}, o_{16}, p_{21}, q_{22}
Translate(Z_4, u_x) → R_{15}
Translate(A_5, u_x) → S_{15}
Translate(B_5, u_x) → T_{15}
Translate(C_5, u_x) → U_{15}
Polygon(R_{15}, S_{15}, T_{15}, U_{15}) → mnohoúhelník40', r_{19}, s_{19}, t_{20}, u_{19}
Translate(C_5, u_x) → V_{15}
Translate(B_5, u_x) → W_{15}
Translate(D_5, u_x) → Z_{15}
Translate(E_5, u_x) → A_{16}
Polygon(V_{15}, W_{15}, Z_{15}, A_{16}) → mnohoúhelník41', v_{19}, w_{19}, z_{18}, a_{21}
Translate(E_5, u_x) → B_{16}
Translate(D_5, u_x) → C_{16}
Translate(F_5, u_x) → D_{16}
Translate(G_5, u_x) → E_{16}
Polygon(B_{16}, C_{16}, D_{16}, E_{16}) → mnohoúhelník42', b_{22}, c_{21}, d_{20}, e_{20}
Translate(J_5, u_x) → F_{16}
Translate(K_5, u_x) → G_{16}
Translate(L_5, u_x) → H_{16}
Translate(M_5, u_x) → I_{16}
Polygon(F_{16}, G_{16}, H_{16}, I_{16}) → mnohoúhelník43', f_{21}, g_{21}, h_{21}, i_{20}
Translate(M_5, u_x) → J_{16}
Translate(L_5, u_x) → K_{16}
Translate(N_5, u_x) → L_{16}
Translate(O_5, u_x) → M_{16}
Polygon(J_{16}, K_{16}, L_{16}, M_{16}) → mnohoúhelník44', j_{23}, k_{23}, l_{24}, m_{23}
Translate(O_5, u_x) → N_{16}
Translate(N_5, u_x) → O_{16}
Translate(P_5, u_x) → P_{16}
Translate(Q_5, u_x) → Q_{16}
Polygon(N_{16}, O_{16}, P_{16}, Q_{16}) → mnohoúhelník45', n_{20}, o_{17}, p_{22}, q_{23}
Translate(Q_5, u_x) → R_{16}
Translate(P_5, u_x) → S_{16}
Translate(R_5, u_x) → T_{16}
Translate(S_5, u_x) → U_{16}
Polygon(R_{16}, S_{16}, T_{16}, U_{16}) → mnohoúhelník46', r_{20}, s_{20}, t_{21}, u_{20}
Translate(T_5, u_x) → V_{16}
Translate(U_5, u_x) → W_{16}
Translate(V_5, u_x) → Z_{16}
Translate(W_5, u_x) → A_{17}
Polygon(V_{16}, W_{16}, Z_{16}, A_{17}) → mnohoúhelník47', v_{20}, w_{20}, z_{19}, a_{22}
Translate(W_5, u_x) → B_{17}
Translate(V_5, u_x) → C_{17}
Translate(Z_5, u_x) → D_{17}
Translate(A_6, u_x) → E_{17}
Polygon(B_{17}, C_{17}, D_{17}, E_{17}) → mnohoúhelník48', b_{23}, c_{22}, d_{21}, e_{21}
Translate(A_6, u_x) → F_{17}
Translate(Z_5, u_x) → G_{17}
Translate(B_6, u_x) → H_{17}
Translate(C_6, u_x) → I_{17}
Polygon(F_{17}, G_{17}, H_{17}, I_{17}) → mnohoúhelník49', f_{22}, g_{22}, h_{22}, i_{21}
Translate(C_6, u_x) → J_{17}
Translate(B_6, u_x) → K_{17}
Translate(D_6, u_x) → L_{17}
Translate(E_6, u_x) → M_{17}
Polygon(J_{17}, K_{17}, L_{17}, M_{17}) → mnohoúhelník50', j_{24}, k_{24}, l_{25}, m_{24}
Translate(W_4, u_x) → N_{17}
Translate(V_4, u_x) → O_{17}
Translate(O_6, u_x) → P_{17}
Translate(P_6, u_x) → Q_{17}
Polygon(N_{17}, O_{17}, P_{17}, Q_{17}) → mnohoúhelník59', n_{21}, o_{18}, p_{23}, q_{24}
Translate(P_6, u_x) → R_{17}
Translate(O_6, u_x) → S_{17}
Translate(Q_6, u_x) → T_{17}
Translate(R_6, u_x) → U_{17}
Polygon(R_{17}, S_{17}, T_{17}, U_{17}) → mnohoúhelník60', r_{21}, s_{21}, t_{22}, u_{21}
Translate(R_6, u_x) → V_{17}
Translate(Q_6, u_x) → W_{17}
Translate(S_6, u_x) → Z_{17}
Translate(T_6, u_x) → A_{18}
Polygon(V_{17}, W_{17}, Z_{17}, A_{18}) → mnohoúhelník61', v_{21}, w_{21}, z_{20}, a_{23}
Translate(T_6, u_x) → B_{18}
Translate(S_6, u_x) → C_{18}
Translate(U_6, u_x) → D_{18}
Translate(V_6, u_x) → E_{18}
Polygon(B_{18}, C_{18}, D_{18}, E_{18}) → mnohoúhelník62', b_{24}, c_{23}, d_{22}, e_{22}
Translate(V_6, u_x) → F_{18}
Translate(U_6, u_x) → G_{18}
Translate(W_6, u_x) → H_{18}
Translate(Z_6, u_x) → I_{18}
Polygon(F_{18}, G_{18}, H_{18}, I_{18}) → mnohoúhelník63', f_{23}, g_{23}, h_{23}, i_{22}
Translate(G_5, u_x) → J_{18}
Translate(F_5, u_x) → K_{18}
Translate(A_7, u_x) → L_{18}
Translate(B_7, u_x) → M_{18}
Polygon(J_{18}, K_{18}, L_{18}, M_{18}) → mnohoúhelník64', j_{25}, k_{25}, l_{26}, m_{25}
Translate(B_7, u_x) → N_{18}
Translate(A_7, u_x) → O_{18}
Translate(C_7, u_x) → P_{18}
Translate(D_7, u_x) → Q_{18}
Polygon(N_{18}, O_{18}, P_{18}, Q_{18}) → mnohoúhelník65', n_{22}, o_{19}, p_{24}, q_{25}
Translate(D_7, u_x) → R_{18}
Translate(C_7, u_x) → S_{18}
Translate(E_7, u_x) → T_{18}
Translate(F_7, u_x) → U_{18}
Polygon(R_{18}, S_{18}, T_{18}, U_{18}) → mnohoúhelník66', r_{22}, s_{22}, t_{23}, u_{22}
Translate(S_5, u_x) → V_{18}
Translate(R_5, u_x) → W_{18}
Translate(G_7, u_x) → Z_{18}
Translate(H_7, u_x) → A_{19}
Polygon(V_{18}, W_{18}, Z_{18}, A_{19}) → mnohoúhelník67', v_{22}, w_{22}, z_{21}, a_{24}
Translate(F_6, u_x) → B_{19}
Translate(G_6, u_x) → C_{19}
Translate(I_7, u_x) → D_{19}
Translate(J_7, u_x) → E_{19}
Polygon(B_{19}, C_{19}, D_{19}, E_{19}) → mnohoúhelník51', b_{25}, c_{24}, d_{23}, e_{23}
Translate(J_7, u_x) → F_{19}
Translate(I_7, u_x) → G_{19}
Translate(K_7, u_x) → H_{19}
Translate(L_7, u_x) → I_{19}
Polygon(F_{19}, G_{19}, H_{19}, I_{19}) → mnohoúhelník68', f_{24}, g_{24}, h_{24}, i_{23}
Translate(L_7, u_x) → J_{19}
Translate(K_7, u_x) → K_{19}
Translate(M_7, u_x) → L_{19}
Translate(N_7, u_x) → M_{19}
Polygon(J_{19}, K_{19}, L_{19}, M_{19}) → mnohoúhelník69', j_{26}, k_{26}, l_{27}, m_{26}
Translate(O_7, u_x) → N_{19}
Translate(P_7, u_x) → O_{19}
Translate(Q_7, u_x) → P_{19}
Translate(R_7, u_x) → Q_{19}
Polygon(N_{19}, O_{19}, P_{19}, Q_{19}) → mnohoúhelník70', n_{23}, o_{20}, p_{25}, q_{26}
Translate(R_7, u_x) → R_{19}
Translate(Q_7, u_x) → S_{19}
Translate(S_7, u_x) → T_{19}
Translate(T_7, u_x) → U_{19}
Polygon(R_{19}, S_{19}, T_{19}, U_{19}) → mnohoúhelník71', r_{23}, s_{23}, t_{24}, u_{23}
Translate(U_7, u_x) → V_{19}
Translate(V_7, u_x) → W_{19}
Translate(W_7, u_x) → Z_{19}
Translate(Z_7, u_x) → A_{20}
Polygon(V_{19}, W_{19}, Z_{19}, A_{20}) → mnohoúhelník72', v_{23}, w_{23}, z_{22}, a_{25}
Translate(F_{16}, u_x) → B_{20}
Translate(G_{16}, u_x) → C_{20}
Translate(H_{16}, u_x) → D_{20}
Translate(I_{16}, u_x) → E_{20}
Polygon(B_{20}, C_{20}, D_{20}, E_{20}) → mnohoúhelník43'', b_4, c_{25}, d_{24}, e_{24}
Translate(J_{16}, u_x) → F_{20}
Translate(K_{16}, u_x) → G_{20}
Translate(L_{16}, u_x) → H_{20}
Translate(M_{16}, u_x) → I_{20}
Polygon(F_{20}, G_{20}, H_{20}, I_{20}) → mnohoúhelník44'', f_{25}, g_{25}, h_{25}, i_{24}
Translate(N_{16}, u_x) → J_{20}
Translate(O_{16}, u_x) → K_{20}
Translate(P_{16}, u_x) → L_{20}
Translate(Q_{16}, u_x) → M_{20}
Polygon(J_{20}, K_{20}, L_{20}, M_{20}) → mnohoúhelník45'', j_{27}, k_{27}, l_{28}, m_{27}
Translate(R_{16}, u_x) → N_{20}
Translate(S_{16}, u_x) → O_{20}
Translate(T_{16}, u_x) → P_{20}
Translate(U_{16}, u_x) → Q_{20}
Polygon(N_{20}, O_{20}, P_{20}, Q_{20}) → mnohoúhelník46'', n_{24}, o_{21}, p_{26}, q_{27}
Translate(V_{16}, u_x) → R_{20}
Translate(W_{16}, u_x) → S_{20}
Translate(Z_{16}, u_x) → T_{20}
Translate(A_{17}, u_x) → U_{20}
Polygon(R_{20}, S_{20}, T_{20}, U_{20}) → mnohoúhelník47'', r_{24}, s_{24}, t_{25}, u_{12}
Translate(B_{17}, u_x) → V_{20}
Translate(C_{17}, u_x) → W_{20}
Translate(D_{17}, u_x) → Z_{20}
Translate(E_{17}, u_x) → A_{21}
Polygon(V_{20}, W_{20}, Z_{20}, A_{21}) → mnohoúhelník48'', v_{24}, w_{12}, z_{11}, a_4
Translate(F_{17}, u_x) → B_{21}
Translate(G_{17}, u_x) → C_{21}
Translate(H_{17}, u_x) → D_{21}
Translate(I_{17}, u_x) → E_{21}
Polygon(B_{21}, C_{21}, D_{21}, E_{21}) → mnohoúhelník49'', b_{26}, c_{26}, d_{25}, e_{25}
Translate(J_{17}, u_x) → F_{21}
Translate(K_{17}, u_x) → G_{21}
Translate(L_{17}, u_x) → H_{21}
Translate(M_{17}, u_x) → I_{21}
Polygon(F_{21}, G_{21}, H_{21}, I_{21}) → mnohoúhelník50'', f_{26}, g_{26}, h_{26}, i_{25}
Translate(V_{18}, u_x) → J_{21}
Translate(W_{18}, u_x) → K_{21}
Translate(Z_{18}, u_x) → L_{21}
Translate(A_{19}, u_x) → M_{21}
Polygon(J_{21}, K_{21}, L_{21}, M_{21}) → mnohoúhelník67'', j_{28}, k_{28}, l_{29}, m_{28}
Translate(B_{19}, u_x) → N_{21}
Translate(C_{19}, u_x) → O_{21}
Translate(D_{19}, u_x) → P_{21}
Translate(E_{19}, u_x) → Q_{21}
Polygon(N_{21}, O_{21}, P_{21}, Q_{21}) → mnohoúhelník51'', n_{25}, o_{22}, p_{27}, q_{28}
Translate(F_{19}, u_x) → R_{21}
Translate(G_{19}, u_x) → S_{21}
Translate(H_{19}, u_x) → T_{21}
Translate(I_{19}, u_x) → U_{21}
Polygon(R_{21}, S_{21}, T_{21}, U_{21}) → mnohoúhelník68'', r_{25}, s_{25}, t_{26}, u_{13}
Translate(J_{19}, u_x) → V_{21}
Translate(K_{19}, u_x) → W_{21}
Translate(L_{19}, u_x) → Z_{21}
Translate(M_{19}, u_x) → A_{22}
Polygon(V_{21}, W_{21}, Z_{21}, A_{22}) → mnohoúhelník69'', v_{25}, w_{24}, z_{12}, a_{26}
Translate(N_{19}, u_x) → B_{22}
Translate(O_{19}, u_x) → C_{22}
Translate(P_{19}, u_x) → D_{22}
Translate(Q_{19}, u_x) → E_{22}
Polygon(B_{22}, C_{22}, D_{22}, E_{22}) → mnohoúhelník70'', b_{27}, c_{27}, d_{26}, e_{26}
Translate(R_{19}, u_x) → F_{22}
Translate(S_{19}, u_x) → G_{22}
Translate(T_{19}, u_x) → H_{22}
Translate(U_{19}, u_x) → I_{22}
Polygon(F_{22}, G_{22}, H_{22}, I_{22}) → mnohoúhelník71'', f_{27}, g_{27}, h_{27}, i_{26}
Translate(V_{19}, u_x) → J_{22}
Translate(W_{19}, u_x) → K_{22}
Translate(Z_{19}, u_x) → L_{22}
Translate(A_{20}, u_x) → M_{22}
Polygon(J_{22}, K_{22}, L_{22}, M_{22}) → mnohoúhelník72'', j_{29}, k_{29}, l_{30}, m_{29}
Translate(J_{15}, u_x) → N_{22}
Translate(K_{15}, u_x) → O_{22}
Translate(L_{15}, u_x) → P_{22}
Translate(M_{15}, u_x) → Q_{22}
Polygon(N_{22}, O_{22}, P_{22}, Q_{22}) → mnohoúhelník38'', n_{26}, o_{23}, p_{28}, q_{29}
Translate(N_{15}, u_x) → R_{22}
Translate(O_{15}, u_x) → S_{22}
Translate(P_{15}, u_x) → T_{22}
Translate(Q_{15}, u_x) → U_{22}
Polygon(R_{22}, S_{22}, T_{22}, U_{22}) → mnohoúhelník39'', r_{26}, s_{26}, t_{27}, u_{24}
Translate(R_{15}, u_x) → V_{22}
Translate(S_{15}, u_x) → W_{22}
Translate(T_{15}, u_x) → Z_{22}
Translate(U_{15}, u_x) → A_{23}
Polygon(V_{22}, W_{22}, Z_{22}, A_{23}) → mnohoúhelník40'', v_{26}, w_{25}, z_{23}, a_{27}
Translate(V_{15}, u_x) → B_{23}
Translate(W_{15}, u_x) → C_{23}
Translate(Z_{15}, u_x) → D_{23}
Translate(A_{16}, u_x) → E_{23}
Polygon(B_{23}, C_{23}, D_{23}, E_{23}) → mnohoúhelník41'', b_{28}, c_{28}, d_{27}, e_{27}
Translate(B_{16}, u_x) → F_{23}
Translate(C_{16}, u_x) → G_{23}
Translate(D_{16}, u_x) → H_{23}
Translate(E_{16}, u_x) → I_{23}
Polygon(F_{23}, G_{23}, H_{23}, I_{23}) → mnohoúhelník42'', f_{28}, g_{28}, h_{28}, i_{27}
Translate(N_{17}, u_x) → J_{23}
Translate(O_{17}, u_x) → K_{23}
Translate(P_{17}, u_x) → L_{23}
Translate(Q_{17}, u_x) → M_{23}
Polygon(J_{23}, K_{23}, L_{23}, M_{23}) → mnohoúhelník59'', j_{30}, k_{30}, l_{31}, m_{30}
Translate(R_{17}, u_x) → N_{23}
Translate(S_{17}, u_x) → O_{23}
Translate(T_{17}, u_x) → P_{23}
Translate(U_{17}, u_x) → Q_{23}
Polygon(N_{23}, O_{23}, P_{23}, Q_{23}) → mnohoúhelník60'', n_{27}, o_{24}, p_{29}, q_{30}
Translate(V_{17}, u_x) → R_{23}
Translate(W_{17}, u_x) → S_{23}
Translate(Z_{17}, u_x) → T_{23}
Translate(A_{18}, u_x) → U_{23}
Polygon(R_{23}, S_{23}, T_{23}, U_{23}) → mnohoúhelník61'', r_{27}, s_{27}, t_{28}, u_{25}
Translate(B_{18}, u_x) → V_{23}
Translate(C_{18}, u_x) → W_{23}
Translate(D_{18}, u_x) → Z_{23}
Translate(E_{18}, u_x) → A_{24}
Polygon(V_{23}, W_{23}, Z_{23}, A_{24}) → mnohoúhelník62'', v_{27}, w_{26}, z_{24}, a_{28}
Translate(F_{18}, u_x) → B_{24}
Translate(G_{18}, u_x) → C_{24}
Translate(H_{18}, u_x) → D_{24}
Translate(I_{18}, u_x) → E_{24}
Polygon(B_{24}, C_{24}, D_{24}, E_{24}) → mnohoúhelník63'', b_{29}, c_{29}, d_{28}, e_{28}
Translate(J_{18}, u_x) → F_{24}
Translate(K_{18}, u_x) → G_{24}
Translate(L_{18}, u_x) → H_{24}
Translate(M_{18}, u_x) → I_{24}
Polygon(F_{24}, G_{24}, H_{24}, I_{24}) → mnohoúhelník64'', f_{29}, g_{29}, h_{29}, i_{28}
Translate(N_{18}, u_x) → J_{24}
Translate(O_{18}, u_x) → K_{24}
Translate(P_{18}, u_x) → L_{24}
Translate(Q_{18}, u_x) → M_{24}
Polygon(J_{24}, K_{24}, L_{24}, M_{24}) → mnohoúhelník65'', j_{31}, k_{31}, l_{32}, m_{31}
Translate(R_{18}, u_x) → N_{24}
Translate(S_{18}, u_x) → O_{24}
Translate(T_{18}, u_x) → P_{24}
Translate(U_{18}, u_x) → Q_{24}
Polygon(N_{24}, O_{24}, P_{24}, Q_{24}) → mnohoúhelník66'', n_{28}, o_{25}, p_{30}, q_{31}
Translate(N_{21}, u_x) → R_{24}
Translate(O_{21}, u_x) → S_{24}
Translate(P_{21}, u_x) → T_{24}
Translate(Q_{21}, u_x) → U_{24}
Polygon(R_{24}, S_{24}, T_{24}, U_{24}) → mnohoúhelník51''', r_{28}, s_{28}, t_{29}, u_{26}
Translate(R_{21}, u_x) → V_{24}
Translate(S_{21}, u_x) → W_{24}
Translate(T_{21}, u_x) → Z_{24}
Translate(U_{21}, u_x) → A_{25}
Polygon(V_{24}, W_{24}, Z_{24}, A_{25}) → mnohoúhelník68''', v_{28}, w_{27}, z_{25}, a_{29}
Translate(V_{21}, u_x) → B_{25}
Translate(W_{21}, u_x) → C_{25}
Translate(Z_{21}, u_x) → D_{25}
Translate(A_{22}, u_x) → E_{25}
Polygon(B_{25}, C_{25}, D_{25}, E_{25}) → mnohoúhelník69''', b_{30}, c_{30}, d_{29}, e_{29}
Translate(B_{22}, u_x) → F_{25}
Translate(C_{22}, u_x) → G_{25}
Translate(D_{22}, u_x) → H_{25}
Translate(E_{22}, u_x) → I_{25}
Polygon(F_{25}, G_{25}, H_{25}, I_{25}) → mnohoúhelník70''', f_{30}, g_{30}, h_{30}, i_{29}
Translate(F_{22}, u_x) → J_{25}
Translate(G_{22}, u_x) → K_{25}
Translate(H_{22}, u_x) → L_{25}
Translate(I_{22}, u_x) → M_{25}
Polygon(J_{25}, K_{25}, L_{25}, M_{25}) → mnohoúhelník71''', j_{32}, k_{32}, l_{33}, m_{32}
Translate(J_{22}, u_x) → N_{25}
Translate(K_{22}, u_x) → O_{25}
Translate(L_{22}, u_x) → P_{25}
Translate(M_{22}, u_x) → Q_{25}
Polygon(N_{25}, O_{25}, P_{25}, Q_{25}) → mnohoúhelník72''', n_{29}, o_{26}, p_{31}, q_{32}
Translate(B_{20}, u_x) → R_{25}
Translate(C_{20}, u_x) → S_{25}
Translate(D_{20}, u_x) → T_{25}
Translate(E_{20}, u_x) → U_{25}
Polygon(R_{25}, S_{25}, T_{25}, U_{25}) → mnohoúhelník43''', r_{29}, s_{29}, t_{30}, u_{27}
Translate(F_{20}, u_x) → V_{25}
Translate(G_{20}, u_x) → W_{25}
Translate(H_{20}, u_x) → Z_{25}
Translate(I_{20}, u_x) → A_{26}
Polygon(V_{25}, W_{25}, Z_{25}, A_{26}) → mnohoúhelník44''', v_{29}, w_{28}, z_{26}, a_{30}
Translate(J_{20}, u_x) → B_{26}
Translate(K_{20}, u_x) → C_{26}
Translate(L_{20}, u_x) → D_{26}
Translate(M_{20}, u_x) → E_{26}
Polygon(B_{26}, C_{26}, D_{26}, E_{26}) → mnohoúhelník45''', b_{31}, c_{31}, d_{30}, e_{30}
Translate(N_{20}, u_x) → F_{26}
Translate(O_{20}, u_x) → G_{26}
Translate(P_{20}, u_x) → H_{26}
Translate(Q_{20}, u_x) → I_{26}
Polygon(F_{26}, G_{26}, H_{26}, I_{26}) → mnohoúhelník46''', f_{31}, g_{31}, h_{31}, i_{30}
Translate(R_{20}, u_x) → J_{26}
Translate(S_{20}, u_x) → K_{26}
Translate(T_{20}, u_x) → L_{26}
Translate(U_{20}, u_x) → M_{26}
Polygon(J_{26}, K_{26}, L_{26}, M_{26}) → mnohoúhelník47''', j_{33}, k_{33}, l_{34}, m_{33}
Translate(V_{20}, u_x) → N_{26}
Translate(W_{20}, u_x) → O_{26}
Translate(Z_{20}, u_x) → P_{26}
Translate(A_{21}, u_x) → Q_{26}
Polygon(N_{26}, O_{26}, P_{26}, Q_{26}) → mnohoúhelník48''', n_{30}, o_{27}, p_{32}, q_{33}
Translate(B_{21}, u_x) → R_{26}
Translate(C_{21}, u_x) → S_{26}
Translate(D_{21}, u_x) → T_{26}
Translate(E_{21}, u_x) → U_{26}
Polygon(R_{26}, S_{26}, T_{26}, U_{26}) → mnohoúhelník49''', r_{30}, s_{30}, t_{31}, u_{28}
Translate(F_{21}, u_x) → V_{26}
Translate(G_{21}, u_x) → W_{26}
Translate(H_{21}, u_x) → Z_{26}
Translate(I_{21}, u_x) → A_{27}
Polygon(V_{26}, W_{26}, Z_{26}, A_{27}) → mnohoúhelník50''', v_{30}, w_{29}, z_{27}, a_{31}
Translate(J_{21}, u_x) → B_{27}
Translate(K_{21}, u_x) → C_{27}
Translate(L_{21}, u_x) → D_{27}
Translate(M_{21}, u_x) → E_{27}
Polygon(B_{27}, C_{27}, D_{27}, E_{27}) → mnohoúhelník67''', b_{32}, c_{32}, d_{31}, e_{31}
Translate(V_{22}, u_x) → F_{27}
Translate(W_{22}, u_x) → G_{27}
Translate(Z_{22}, u_x) → H_{27}
Translate(A_{23}, u_x) → I_{27}
Polygon(F_{27}, G_{27}, H_{27}, I_{27}) → mnohoúhelník40''', f_{32}, g_{32}, h_{32}, i_{31}
Translate(B_{23}, u_x) → J_{27}
Translate(C_{23}, u_x) → K_{27}
Translate(D_{23}, u_x) → L_{27}
Translate(E_{23}, u_x) → M_{27}
Polygon(J_{27}, K_{27}, L_{27}, M_{27}) → mnohoúhelník41''', j_{34}, k_{34}, l_{35}, m_{34}
Translate(F_{23}, u_x) → N_{27}
Translate(G_{23}, u_x) → O_{27}
Translate(H_{23}, u_x) → P_{27}
Translate(I_{23}, u_x) → Q_{27}
Polygon(N_{27}, O_{27}, P_{27}, Q_{27}) → mnohoúhelník42''', n_{31}, o_{28}, p_{33}, q_{34}
Translate(F_{24}, u_x) → R_{27}
Translate(G_{24}, u_x) → S_{27}
Translate(H_{24}, u_x) → T_{27}
Translate(I_{24}, u_x) → U_{27}
Polygon(R_{27}, S_{27}, T_{27}, U_{27}) → mnohoúhelník64''', r_{31}, s_{31}, t_{32}, u_{29}
Translate(J_{24}, u_x) → V_{27}
Translate(K_{24}, u_x) → W_{27}
Translate(L_{24}, u_x) → Z_{27}
Translate(M_{24}, u_x) → A_{28}
Polygon(V_{27}, W_{27}, Z_{27}, A_{28}) → mnohoúhelník65''', v_{31}, w_{30}, z_{28}, a_{32}
Translate(N_{24}, u_x) → B_{28}
Translate(O_{24}, u_x) → C_{28}
Translate(P_{24}, u_x) → D_{28}
Translate(Q_{24}, u_x) → E_{28}
Polygon(B_{28}, C_{28}, D_{28}, E_{28}) → mnohoúhelník66''', b_{33}, c_{33}, d_{32}, e_{32}
Translate(N_{22}, u_x) → F_{28}
Translate(O_{22}, u_x) → G_{28}
Translate(P_{22}, u_x) → H_{28}
Translate(Q_{22}, u_x) → I_{28}
Polygon(F_{28}, G_{28}, H_{28}, I_{28}) → mnohoúhelník38''', f_{33}, g_{33}, h_{33}, i_{32}
Translate(R_{22}, u_x) → J_{28}
Translate(S_{22}, u_x) → K_{28}
Translate(T_{22}, u_x) → L_{28}
Translate(U_{22}, u_x) → M_{28}
Polygon(J_{28}, K_{28}, L_{28}, M_{28}) → mnohoúhelník39''', j_{35}, k_{35}, l_{36}, m_{35}
Translate(J_{23}, u_x) → N_{28}
Translate(K_{23}, u_x) → O_{28}
Translate(L_{23}, u_x) → P_{28}
Translate(M_{23}, u_x) → Q_{28}
Polygon(N_{28}, O_{28}, P_{28}, Q_{28}) → mnohoúhelník59''', n_{32}, o_{29}, p_{34}, q_{35}
Translate(N_{23}, u_x) → R_{28}
Translate(O_{23}, u_x) → S_{28}
Translate(P_{23}, u_x) → T_{28}
Translate(Q_{23}, u_x) → U_{28}
Polygon(R_{28}, S_{28}, T_{28}, U_{28}) → mnohoúhelník60''', r_{32}, s_{32}, t_{33}, u_{30}
Translate(R_{23}, u_x) → V_{28}
Translate(S_{23}, u_x) → W_{28}
Translate(T_{23}, u_x) → Z_{28}
Translate(U_{23}, u_x) → A_{29}
Polygon(V_{28}, W_{28}, Z_{28}, A_{29}) → mnohoúhelník61''', v_{32}, w_{31}, z_{29}, a_{33}
Translate(V_{23}, u_x) → B_{29}
Translate(W_{23}, u_x) → C_{29}
Translate(Z_{23}, u_x) → D_{29}
Translate(A_{24}, u_x) → E_{29}
Polygon(B_{29}, C_{29}, D_{29}, E_{29}) → mnohoúhelník62''', b_{34}, c_{34}, d_{33}, e_{33}
Translate(B_{24}, u_x) → F_{29}
Translate(C_{24}, u_x) → G_{29}
Translate(D_{24}, u_x) → H_{29}
Translate(E_{24}, u_x) → I_{29}
Polygon(F_{29}, G_{29}, H_{29}, I_{29}) → mnohoúhelník63''', f_{34}, g_{34}, h_{34}, i_{33}

AI Drawing Prompt Reference

Start at the origin with horizontal and vertical axes. Draw a circle centered at the origin intersecting the horizontal axis at a point, then draw a vertical line through that point. Similarly, intersect the vertical axis and draw a horizontal line. Using these points as centers, repeatedly draw circles to generate equally spaced points along both axes. Connect corresponding points with parallel lines to form a grid, illustrating the geometric derivation of the sum of the first n natural numbers.

The prompt above was compiled from the original drawing commands and is for reference only.