Los triángulos AHB y AHC son semejantes

c

n

m

b

a

h

m

h

h

n

\frac{h}{n}=\frac{m}{h}⇒h^2=m·n⇒h=\sqrt{m·n}

Teorema de la altura

Teorema del cateto

Teorema de la altura

Teorema del cateto

Los triángulos ABC y ACH son semejantes

m

a

b

b

\frac{b}{a}=\frac{m}{b}⇒b^2=a·m⇒b=\sqrt{a·m}

b

Comprobación

Mueve los puntos A y B

Teorema de la altura

Teorema del cateto

Comprobación

Mueve los puntos A y B

B1 = (1, 2)
B2 = (5, 2)
a = Segment(B1, B2)
C = (-3, 2)
B = Point(a)
c = Semicircle(C, B)
A_1 = Midpoint(C, B)
b = OrthogonalLine(A_1, a)
D = Intersect(c, b, 1)
d = CircleArc(A_1, D, C)
A = Point(d)
poli1, b_1, c_1, e = Polygon(C, A, B)
f = OrthogonalLine(A, e)
H = Intersect(f, e)
poli2, a_1, c_2, h = Polygon(C, H, A)
poli3, b_2, a_2, h_1 = Polygon(A, H, B)
u = Vector(A, H)
E1 = (-4, -4)
E5 = (0, -4)
g = Segment(E1, E5)
E2: E1 + (E5 - E1) / 4
E3 = Mirror(E1, E2)
E4 = Mirror(E2, E3)
E = Point(g)
e1: x(E) - x(E1)
e2: x(E) - x(E2)
e3: x(E) - x(E3)
e4: x(E) - x(E4)
f2 = If(e2 < 0, 0, 0 ≤ e2 ≤ 1, e2, 1)
f1 = If(e1 < 0, 0, 0 ≤ e1 ≤ 1, e1, 1)
f3 = If(e3 < 0, 0, 0 ≤ e3 ≤ 1, e3, 1)
f4 = If(e4 < 0, 0, 0 ≤ e4 ≤ 1, e4, 1)
t = 1.5 (slider from 1 to 2, step 0.05)
A' = Translate(A, Vector[((f1 * t) * u)])
H' = Translate(H, Vector[((f1 * t) * u)])
B' = Translate(B, Vector[((f1 * t) * u)])
poli3', b', a', h' = Polygon(A', H', B')
C' = Translate(C, Vector[((f2 * t) * u)])
H'_1 = Translate(H, Vector[((f2 * t) * u)])
A'_1 = Translate(A, Vector[((f2 * t) * u)])
poli2', a'_1, c', h'_1 = Polygon(C', H'_1, A'_1)
F = Rotate(H'_1, ((-((f4 * 90))))°, H')
C'' = Rotate(C', (-(((f3 * 90))°)), H')
G = Rotate(H'_1, (-(((f3 * 90))°)), H')
I = Rotate(A'_1, (-(((f3 * 90))°)), H')
poli2'', i, c'', g_1 = Polygon(C'', G, I)
v = Vector(I, B')
C''' = Translate(C'', Vector[(f4 * v)])
G' = Translate(G, Vector[(f4 * v)])
I' = Translate(I, Vector[(f4 * v)])
poli2''', i', c''', g' = Polygon(C''', G', I')
j: false
th: 1
k2 = Segment(C, H)
k3 = Segment(H, B)
J = Point(g)
k: false
C'_1 = Translate(C, Vector[((u * t) * f1)])
A'_2 = Translate(A, Vector[((u * t) * f1)])
B'_1 = Translate(B, Vector[((u * t) * f1)])
j1: x(J) - x(E1)
j2: x(J) - x(E2)
j3: x(J) - x(E3)
j4: x(J) - x(E4)
h1 = If(j1 < 0, 0, 0 ≤ j1 ≤ 1, j1, 1)
h2 = If(j2 < 0, 0, 0 ≤ j2 ≤ 1, j2, 1)
h3 = If(j3 < 0, 0, 0 ≤ j3 ≤ 1, j3, 1)
h4 = If(j4 < 0, 0, 0 ≤ j4 ≤ 1, j4, 1)
C'_2 = Translate(C, Vector[((t * h1) * u)])
A'_3 = Translate(A, Vector[((t * h1) * u)])
B'_2 = Translate(B, Vector[((t * h1) * u)])
poli1', b'_2, c'_2, a'_3 = Polygon(C'_2, A'_3, B'_2)
polígono1, b'_1, c'_1, a'_2 = Polygon(C'_2, A'_3, B'_2)
p = Circle(B'_2, h)
q = Circle(B'_2, c_2)
K = Intersect(q, c'_1, 1)
L = Intersect(p, a'_2, 1)
C'_3 = Translate(C, Vector[((h2 * t) * u)])
H'_2 = Translate(H, Vector[((h2 * t) * u)])
A'_4 = Translate(A, Vector[((h2 * t) * u)])
poli2'_1, a'_4, c'_3, h'_2 = Polygon(C'_3, H'_2, A'_4)
l = Line(A, H)
M = Mirror(C'_3, l)
N = Mirror(H'_2, l)
O = Mirror(A'_4, l)
poli4, o, m, n = Polygon(M, N, O)
r = LineBisector(c'_1)
s = LineBisector(K, H'_2)
P = Intersect(r, s)
α = Angle(L, P, M)
M' = Rotate(M, ((-α) * h4), P)
N' = Rotate(N, ((-α) * h4), P)
O' = Rotate(O, ((-α) * h4), P)
poli4', o', m', n' = Polygon(M', N', O')
t1: x(E1) < x(J) ≤ x(E2)
t2: x(E2) < x(J) ≤ x(E3)
t3: x(E3) < x(J) ≤ x(E4)
t4: x(E4) < x(J) ≤ x(E5)
Q = Point(k2)
d_1 = Segment(A, H)
w: false
e_1: (k2 * k3)
f_1: sqrt(e_1)
texto23: "h=\sqrt{" + (LaTeX[k2]) + "·" + (LaTeX[k3]) + "}=\sqrt{" + (LaTeX[e_1]) + "}=" + (LaTeX[f_1]) + ""
texto21: "n=" + (LaTeX[k3]) + ""
texto22: "m=" + (LaTeX[k2]) + ""
texto20: "h=" + (LaTeX[d_1]) + " "
i_1: d_1 / k3
texto25: "\frac{h}{n}=\frac{" + (LaTeX[d_1]) + "}{" + (LaTeX[k3]) + "}=" + (LaTeX[i_1]) + ""
j_1: k2 / d_1
texto13: "\frac{m}{h}=\frac{" + (LaTeX[k2]) + "}{" + (LaTeX[d_1]) + "}=" + (LaTeX[j_1]) + ""
k_1: false
p1 = Distance(A, C)
r1 = Distance(C, B)
q1 = Distance(C, H)
texto18: "a=" + (LaTeX[r1]) + ";\:b=" + (LaTeX[p1]) + ";\:m=" + (LaTeX[q1]) + ""
s1: p1 / r1
texto28: "\frac{b}{a}=\frac{" + (LaTeX[p1]) + "}{" + (LaTeX[r1]) + "}=" + (LaTeX[s1]) + ""
l_1: q1 / p1
texto29: "\frac{m}{b}=\frac{" + (LaTeX[q1]) + "}{" + (LaTeX[p1]) + "}=" + (LaTeX[l_1]) + ""
m_1: sqrt((r1 * q1))
texto30: "b=\sqrt{" + (LaTeX[r1]) + "·" + (LaTeX[q1]) + "}=" + (LaTeX[m_1]) + ""