Criterio de semejanza

1

2

3

1. Dos triángulos son semejantes si tienen
    dos ángulos iguales.

A'

A'

B'

C'

\widehat{A}=\widehat{A'} \:;\widehat{B}=\widehat{B'}⇒\widehat{C}=\widehat{C'}

Los\:triángulos\:ABC\:y \:A'B'C'\:están \:en\: posición\: de\: Tales,

⇒ Los \:Triángulos\:ABC\:y\: A'B'C'\:son \:semejantes.

\frac{A'B'}{AB}=\frac{B'C'}{BC}=\frac{A'C'}{AC}

Mueve los puntos para fijar los ángulos A y B

1

2

3

2. Dos triángulos son semejantes
     si tienen un ángulo igual y los lados
     que lo forman son proporcionales.

3. Dos triángulos son semejantes
     si tienen los tres lados proporcionales.

A

B

C

Los lados a,b y c no forman triángulo.

A'

A'

C'

B'

Los\:triángulos\:ABC\:y \:A'B'C' \:están \:en \:posición\: de\: Tales

\widehat{A}=\widehat{A'},\widehat{B}=\widehat{B'}, \widehat{C}=\widehat{C'}.

Los\: triángulos \:ABC\:y \:A'B'C'\:son\:semejantes.

A

AC

AB

C

B

A'

A'

C'

B'

Los\: triángulos\:ABC\:y \:A'B'C' están\:en\:posición\:de\:Tales.

Los\:triángulos \:ABC\:y \:A'B'C' \:son \:semejantes\:⇔

\frac{A'B'}{AB}=\frac{A'C'}{AC};\:\: \widehat{A}=\widehat{A'}

⇔\frac{A'B'}{AB}=\frac{A'C'}{AC}=\frac{B'C'}{BC};\:\: \widehat{A}=\widehat{A'};\widehat{B}=\widehat{B'};\:\widehat{C}=\widehat{C'}

cri: 1
A = (-2.04, 1)
a = Line(A, xAxis)
B_1 = Point(a)
b = Ray(A, B_1)
r1 = 1 (slider from 0.5 to 1, step 0.05)
C_1 = (-3, 1)
c = Circle(C_1, r1)
D = (0, 1)
d = Circle(D, r1)
e = Line(C_1, xAxis)
E = Intersect(c, e, 1)
F = Intersect(d, e, 2)
G = Intersect(c, e, 2)
H = Intersect(d, e, 1)
f = Semicircle(E, G)
g = Semicircle(H, F)
I = Point(f)
α = Angle(G, C_1, I)
h = Segment(C_1, I)
i = Line(A, h)
J = Point(b)
j = Segment(A, J)
B = Point(j)
k = Line(D, h)
L = Intersect(g, k, 1)
p = CircleArc(D, L, H)
M = Point(p)
l = Segment(M, D)
m = Line(B, l)
C = Intersect(i, m)
polígono1, n, k_1, a_1 = Polygon(B, A, C)
β = Angle(B, A, C)
γ = Angle(M, D, H)
δ = Angle(C, B, A)
q = Segment(A, B_1)
O = Point(q)
r = Segment(O, B_1)
s = Line(O, k_1)
t = 1.4 (slider from 0.5 to 2, step 0.05)
P: O + (t * (B - A))
b_1 = Line(O, k_1)
c_1 = Line(P, a_1)
Q = Intersect(b_1, c_1)
tri2, q_1, o, p_1 = Polygon(O, P, Q)
u = Vector(O, A)
t1: 0
O' = Translate(O, Vector[(t1 * u)])
P' = Translate(P, Vector[(t1 * u)])
Q' = Translate(Q, Vector[(t1 * u)])
tri2', q', o', p' = Polygon(O', P', Q')
ε = Angle(P', O', Q')
ζ = Angle(Q', P', O')
K = (-2, 1.5)
A1 = (-3.5, 2)
A3 = (1.5, 2)
d_1 = Segment(A1, A3)
B1 = (-3.5, 1)
B3 = (1.5, 1)
e_1 = Segment(B1, B3)
C1 = (-3.5, 0)
C3 = (1.5, 0)
f_1 = Segment(C1, C3)
A2 = Point(d_1)
B2 = Point(e_1)
C2 = Point(f_1)
g_1 = Segment(A1, A2)
h_1 = Segment(B1, B2)
i_1 = Segment(C1, C2)
r_1 = Circle(K, i_1)
j_1 = Line(K, xAxis)
N = Intersect(r_1, j_1, 2)
s_1 = Circle(K, g_1)
t_1 = Circle(N, h_1)
R = Intersect(s_1, t_1, 1)
g_2 = Segment(K, R)
h_2 = Segment(R, N)
i_2 = Segment(K, N)
polígono2, n_2, k_2, r_2 = Polygon(K, R, N)
v1: g_1 + h_1 > i_1
v2: h_1 + i_1 > g_1
v3: i_1 + g_1 > h_1
v: v1 ∧ v2 ∧ v3
razón = 1.55 (slider from 0.5 to 2, step 0.05)
S = (2, -1)
a_2 = Line(S, g_2)
c_2 = Circle(S, (razón * g_1))
w = Vector(S, K)
t3: 0
T: S + (t3 * (K - S))
l_1 = Line(T, g_2)
m_1 = Line(T, i_2)
d_2 = Circle(T, (razón * g_2))
U = Intersect(d_2, l_1, 2)
e_2 = Circle(T, (razón * i_2))
n_1 = Line(T, i_2)
V = Intersect(e_2, n_1, 2)
polígono3, g_3, h_3, i_3 = Polygon(T, U, V)
W = (-4, 1)
Z = (-2, 1)
f_2 = Semicircle(W, Z)
A_1 = Midpoint(W, Z)
D_1 = Point(f_2)
b_2 = Segment(A_1, Z)
j_2 = Segment(A_1, D_1)
E_1 = (-2, 2)
l_2 = Line(E_1, b_2)
m_2 = Line(E_1, j_2)
F_1 = (-4, 0)
G_1 = (1, 0)
p_2 = Segment(F_1, G_1)
H_1 = (-4, -1)
I_1 = (1, -1)
q_2 = Segment(H_1, I_1)
J_1 = Point(p_2)
K_1 = Point(q_2)
s_2 = Segment(F_1, J_1)
t_2 = Segment(H_1, K_1)
c_3 = Circle(E_1, s_2)
L_1 = Intersect(c_3, m_2, 2)
d_3 = Circle(E_1, t_2)
M_1 = Intersect(d_3, l_2, 2)
polígono4, m_3, e_3, l_3 = Polygon(E_1, L_1, M_1)
η = Angle(Z, A_1, D_1)
raz2 = 1.6 (slider from 0.5 to 2, step 0.1)
N_1 = (2, -1)
t2: 0
O_1: N_1 + (t2 * (E_1 - N_1))
a_3 = Line(O_1, j_2)
b_3 = Line(O_1, b_2)
f_3 = Circle(O_1, (s_2 * raz2))
k_3 = Circle(O_1, (t_2 * raz2))
P_1 = Intersect(f_3, a_3, 2)
Q_1 = Intersect(k_3, b_3, 2)
polígono5, q_3, o_1, p_3 = Polygon(O_1, P_1, Q_1)
θ = Angle(M_1, E_1, L_1)
ι = Angle(Q_1, O_1, P_1)
texto19: "Proporción lados=" + razón + ""
texto20: "Proporción lados=" + raz2 + ""
texto7: "Proporción = " + t + ""