RELACIONES EN TRIÁNGULOS RECTÁNGULOS
Author: Miriam y Nerea

Mueve el deslizador y verás que se siguen verificando las relaciones.

 Triángulo AHC : AC² = AH² + CH² 

Triángulo AHB : AB² = AH² + HB²

Triángulo ABC : BC² = AC² + AB²

A

B

H

C

H

Entre el triángulo AHC y AHB :

  \frac{ AC }{ AB }=          \frac{ HB }{ AH }=       \frac{ AH }{ HC }  

Entre el triángulo ABC y AHB :

    \frac{ BC }{ AC }=          \frac{ AB }{ AH }=       \frac{ HC }{ AC }  

Entre el triángulo ABC y AHB :

  \frac{ BC }{ AB }=          \frac{ AB }{ HB }=       \frac{ AC }{ AH }  

BC

AB

AC

Pitágoras

Tales

Superponer los triángulos por su ángulo recto

Rehacer los triángulos

Relaciones métricas

AH² = HB x HC

BC x AH = AB x AC 

AB²= BC x HB

O = (4.858572167987322, 1.1046925007051254)
b = Line(O, xAxis)
E = Point(b)
a = 5 (slider from -5 to 5, step 0.1)
c = Circle(O, a)
A = Point(c)
t: 0.001
d2 = Segment(A, E)
e2 = Circle(A, If[t  <  d2, t, d2])
C, B = Intersect(c, b)
e = Segment(A, C)
i = OrthogonalLine(E, b)
k = Circle(E, e)
G, I = Intersect(k, i)
b2 = Segment(C, I)
c2 = Circle(C, If[t  <  b2, t, b2])
t1 = Segment(A, E)
h1 = Line(A, E)
f1 = Segment(A, E)
g1 = Circle(A, If[t  <  f1, t, f1])
U, V = Intersect(g1, h1)
n = Line(C, I)
d1 = Segment(C, I)
p = Circle(C, If[t  <  t1, t, t1])
f = Segment(A, B)
h = Circle(E, f)
D, F = Intersect(h, b)
i1 = Segment(B, F)
k1 = Circle(B, If[t  <  i1, t, i1])
d = OrthogonalLine(A, b)
H = Intersect(d, b)
g = Segment(A, H)
j = Segment(I, F)
l = Segment(I, E)
m = Segment(E, F)
q = Line(A, b)
L = Intersect(q, i)
r = Segment(H, B)
s = Circle(E, r)
M, N = Intersect(s, b)
a1 = Segment(L, N)
c1 = Circle(E, g)
J, K = Intersect(c1, b)
b1 = Segment(C, H)
e1 = Circle(E, b1)
P, Q = Intersect(e1, i)
poly1, f_1, i_1, e_1 = Polygon(I, E, F)
poly2, b_1, a_1, c_1 = Polygon(A, C, B)
poly3, b_2, a_2, h_1 = Polygon(A, H, B)
poly4, n_1, l_1, e_2 = Polygon(L, E, N)
poly5, h_2, a_3, c_2 = Polygon(A, C, H)
poly6, k_1, q_1, e_3 = Polygon(Q, E, K)
T = Intersect(f1, h1)
Z, H1 = Intersect(k1, t1)
j1_1 = Segment(A, L)
p1 = Circle(A, If[t  <  j1_1, t, j1_1])
A1, B1 = Intersect(p1, j1_1)
u = Vector(A, A1)
A' = Translate(A, u)
H' = Translate(H, u)
B' = Translate(B, u)
poly3', b', a', h' = Polygon(A', H', B')
m1_2 = Segment(A, K)
q1 = Circle(A, If[t  <  m1_2, t, m1_2])
C1, D1 = Intersect(q1, m1_2)
l1_2 = Segment(C, Q)
r1 = Circle(C, If[t  <  l1_2, t, l1_2])
E1 = Intersect(r1, l1_2, 1)
n1 = Segment(H, E)
s1 = Circle(H, If[t  <  n1, t, n1])
F1, G1 = Intersect(s1, b)
poly8, g1_1, c1_1, e1_1 = Polygon(C1, E1, G1)
j1 = Segment(B, F)
I1, J1 = Intersect(k1, j1)
S, R = Intersect(c2, b2)
W, K1 = Intersect(e2, d2)
l1 = Segment(S, W)
m1 = Segment(W, I1)
a2 = Segment(I1, S)
poly7, i1_1, s_1, w = Polygon(S, W, I1)
f2 = Circle(H, a / 10)
L1 = Intersect(f2, r)
M1 = Intersect(f2, b1)
N1 = Intersect(f2, g)
g2 = Circle(A, a / 10)
O1 = Intersect(g2, f)
P1 = Intersect(g2, e)
h2 = Line(O1, e)
i2 = Line(P1, f)
Q1 = Intersect(i2, h2)
j2 = Segment(O1, Q1)
k2 = Segment(Q1, P1)
l2 = Line(M1, g)
m2 = Line(L1, g)
n2 = Line(N1, r)
R1 = Intersect(l2, n2)
S1 = Intersect(m2, n2)
poly9, r1_1, m1_1, h_3, n1_1 = Polygon(R1, M1, H, N1)
poly10, n1_2, s1_1, l1_1, h_4 = Polygon(N1, S1, L1, H)
p2 = Circle(E, a / 10)
U1 = Intersect(p2, c1_1)
r2 = Line(U1, r)
V1 = Intersect(p2, e1_1)
q2 = Line(V1, g)
s2 = Line(U1, r)
W1 = Intersect(s2, q2)
poly11, u1, w1, v1, e_4 = Polygon(U1, W1, V1, E)
Z1 = Midpoint(w)
A2 = Midpoint(h')
B2 = Midpoint(g1_1)
C2 = (11.165524610437755, -0.6885873884952762)
D2 = (11.145209390271736, -1.8668701581243758)
E2 = (12.811057443885291, -1.8668701581243758)
F2 = (12.811057443885291, -0.6885873884952762)
poly12, c2_1, d2_1, e2_1, f2_1 = Polygon(C2, D2, E2, F2)
G2 = (10.231024482800883, -2.0903375799505843)
H2 = (10.231024482800883, -3.1060985882515326)
I2 = (11.876557316248418, -3.1060985882515326)
J2 = (11.83592687591638, -2.0700223597845655)
poly13, g2_1, h2_1, i2_1, j2_1 = Polygon(G2, H2, I2, J2)
K2 = (13.664296690858087, -2.5982180841010587)
L2 = (14.476905497498846, -2.618533304267078)
v = Vector(K2, L2)
M2 = (13.623666250526048, -1.318359213641864)
N2 = (14.517535937830882, -1.3386744338078829)
z = Vector(M2, N2)
O2 = (13.684611911024108, -3.6952399730660828)
P2 = (14.497220717664867, -3.6952399730660828)
z1 = Vector(O2, P2)
Q2 = (10.231024482800883, -3.329566010077741)
R2 = (10.231024482800883, -4.2843813578806325)
S2 = (11.896872536414437, -4.304696578046651)
U2 = (11.876557316248418, -3.329566010077741)
poly14, q2_1, r2_1, s2_1, u2 = Polygon(Q2, R2, S2, U2)