C:\Users\maria sapuppo\Desktop\Pitagora\Esercizi\dimostrazioni geogebra\Geogebra\16L'Arbelo di Archimedegeogebra-export

Q

S

A

C

P

R

Q

S_1

A_1

A_2

B_1

B_2

A_1

B_1

C_1

A_2

B_2

C_2

C_1

C_2

Avanti

Indietro

reset

Ricordiamo che vale la relazione pitagorica
 anche tra i semicerchi costruiti sui tre lati 
del triangolo rettangolo . Quindi
B≡ B_1 + B_2

La stessa proprietà la applichiamo 
al triangolo PQS_1 , quindi
B_1≡A_1+C_1

Tracciamo il segmento QS_1 ,
 simmetrico di QS rispetto a PR. 
Disegniamo quindi il triangolo che 
ha ipotenusa PR e relativa altezza QS_1

Analogamente , nel triangolo RQS_1
B_2≡ A_2 + C_2

A

B

Dimostriamo che A≡C

Se      B_1+B_2≡ A+A_1+A_2
e         B_1+B_2≡C_1+C_2+A_1+A_2
deduciamo che  
A≡C_1+C_2≡C

S

P = (2, 2)
R = (8, 2)
f = Segment(P, R)
c = Semicircle(P, R)
Q_{1} = Point(f)
d = Semicircle(P, Q_{1})
e = Semicircle(Q_{1}, R)
S_{1} = Point(c)
g = Segment(Q_{1}, S_{1})
A = Point(d)
h = CircumcircleSector(Q_{1}, A, P)
B_{2} = Point(e)
k = CircumcircleSector(R, B_{2}, Q_{1})
p = CircumcircleSector(R, S_{1}, P)
C_{1} = (10, 2)
D = (16, 2)
i = Segment(C_{1}, D)
q = Semicircle(C_{1}, D)
E = Point(i)
S = Point(q)
j = Segment(E, S)
r = Semicircle(C_{1}, E)
s = Semicircle(E, D)
C = Midpoint(j)
t = Circle(C, E)
Q = Intersect(r, s, 1)
F = Point(t)
c_1 = CircumcircleSector(S, F, Q)
G = Point(t)
d_1 = CircumcircleSector(Q, G, S)
H = (2, -3)
I = (8, -3)
l = Segment(H, I)
e_1 = Semicircle(H, I)
J = Point(e_1)
f_1 = CircumcircleSector(H, J, I)
K = (4, -3)
g_1 = Semicircle(H, K)
h_1 = Semicircle(K, I)
m = OrthogonalLine(K, l)
L = Intersect(e_1, f_1, 1)
M = Intersect(e_1, f_1, 1)
N = Intersect(e_1, f_1, 1)
O = Intersect(f_1, m)
O' = Mirror(O, l)
t1, i_1, h_2, o' = Polygon(H, O', I)
k_1 = Semicircle(I, H)
p_1 = Semicircle(O', H)
q_1 = Semicircle(I, O')
T = Point(p_1)
r_1 = CircumcircleSector(O', T, H)
U = Point(q_1)
s_1 = CircumcircleSector(I, U, O')
α = Angle(I, O', H)
n = Segment(K, O')
V = (12, -4)
W = (18, -4)
a = Segment(V, W)
t_1 = Semicircle(V, W)
c_2 = Semicircle(W, V)
Z = Point(a)
b = OrthogonalLine(Z, a)
A_1 = Intersect(c_2, b, 1)
t2, w, v, a_1 = Polygon(V, A_1, W)
β = Angle(V, Z, A_1)
j_1 = Segment(A_1, Z)
l_1 = Segment(V, Z)
m_1 = Segment(Z, A_1)
D_1 = Point(l_1)
B_1 = Point(w)
E_1 = Intersect(a_1, j_1)
d_2 = Semicircle(B_1, E_1)
F_1 = Intersect(c_2, w, 2)
e_2 = Semicircle(E_1, F_1)
f_2 = Semicircle(F_1, B_1)
G_1 = Point(d_2)
g_2 = CircumcircleSector(B_1, G_1, E_1)
H_1 = Point(e_2)
k_2 = CircumcircleSector(E_1, H_1, F_1)
I_1 = Point(f_2)
p_2 = CircumcircleSector(F_1, I_1, B_1)
J_1 = (20, -4)
K_1 = (26, -4)
n_1 = Segment(J_1, K_1)
q_2 = Semicircle(J_1, K_1)
r_2 = Semicircle(K_1, J_1)
L_1 = Point(n_1)
b_1 = OrthogonalLine(L_1, n_1)
M_1 = Intersect(r_2, b_1, 1)
t3, i_2, j_2, l_2 = Polygon(J_1, M_1, K_1)
γ = Angle(M_1, L_1, K_1)
m_2 = Segment(L_1, M_1)
N_1 = Point(n_1)
n_2 = Segment(L_1, N_1)
s_2 = Semicircle(L_1, N_1)
t_2 = Semicircle(N_1, M_1)
c_3 = Semicircle(M_1, L_1)
O_1 = Point(s_2)
d_3 = CircumcircleSector(L_1, O_1, N_1)
P_1 = Point(t_2)
e_3 = CircumcircleSector(N_1, P_1, M_1)
R_1 = Point(c_3)
f_3 = CircumcircleSector(M_1, R_1, L_1)
T_1 = Point(g_1)
U_1 = Point(h_1)
g_3 = CircumcircleSector(N, T_1, K)
h_3 = CircumcircleSector(K, U_1, I)
k_3 = CircumcircleSector(I, U, O')
p_3 = CircumcircleSector(O', T, N)
q_3 = Semicircle(K, O')
r_3 = Semicircle(O', K)
V_1 = Point(q_3)
s_3 = CircumcircleSector(K, V_1, O')
W_1 = Point(r_3)
t_3 = CircumcircleSector(O', W_1, K)
c_4 = CircumcircleSector(P, S_{1}, R)
minimo = 0 (slider from -5 to 5)
o: 0
Z_1 = (20, -4)
A_2 = (26, -4)
tratto1: PolyLine[(21.229,7.9914), (21.229,7.9914), (NaN,NaN), true]
tratto2: PolyLine[(24.357,6.3576), (24.357,6.3576), (NaN,NaN), true]
a_2 = OrthogonalLine(K, l)
B = Intersect(e_1, f_1)
C_2 = Intersect(f_1, a_2)
b_2 = Segment(K, C_2)
massimo = 5 (slider from -5 to 5)