A

B

C

H_A

sen \space C=\frac{h_A}{b} \space ⇒h_A=b \space sen \space C

sen \space  B= \space \frac{h_A}{c}⇒h_A=c \space sen \space B

Teorema del seno

Teorema del coseno

H_C

Comprobación

Altura sobre el lado:

AB

CA

sen \space A=\frac{h_C}{b} \space ⇒h_C=b \space sen \space A

sen \space  B= \space \frac{h_C}{a}⇒h_C=a \space sen \space B

BC

b \space sen \space C=c \space sen \space B⇒\frac{b}{sen \space B}=\frac{c}{sen \space C}

H_B

h_B

b \space sen \space A=a \space sen \space B⇒\frac{a}{sen \space A}=\frac{b}{sen \space B}

sen \space C=\frac{h_B}{a} \space ⇒h_B=a \space sen \space C

sen \space  A= \space \frac{h_B}{c}⇒h_B=c \space sen \space A

c \space sen \space A=a \space sen \space C⇒\frac{c}{sen \space C}=\frac{a}{sen \space A}

h_A

\frac{a}{sen \space A}=\frac{b}{sen B}=\frac{c}{sen \space C}

c

a

b

a-x

x

 h^{2}=b^{2}-x^{2} \space  \space  \space  \space  \space  \space  \space  \space  (1)

c^{2} =h^{2}+\left(a-x   \right)^{2}  \space  \space (2)

Sustituyendo \space valor \space  de \space  h^2 \space de \space  (1) \space en \space (2)

c^{2}=b^{2}-x^{2}+\left(a-x   \right)^2=b^2+a^2-2ax \space   (3)

cos \space C=\frac{x}{b}⇒x=b \space cos \space C \space  \   (4)

Sustituyendo \space  el \space  valor \space  de \space  x  \space de \space  (4) \space  en \space  (3)

c^2=a^2+b^2-2ab \space cos \space C

c^{2} =h^{2}+\left(a+x   \right)^{2}  \space  \space (2)

c^{2}=b^{2}-x^{2}+\left(a+x   \right)^2=b^2+a^2+2ax \space  (3)

cos \space C=\frac{-x}{b}⇒x=-b \space cos \space C \space  \space (4)

a+x

h

 h^{2}=b^{2}-(a+x)^{2} \space  \space  \space  \space  \space  \space  \space  \space   (1)

c^{2} =h^{2}+x^{2}  \space  \space (2)

c^{2}=b^{2}+x^{2}+\left(a+x   \right)^2=b^2-a^2-2ax \space  (3)

cos \space C=\frac{a+x}{b}⇒x=b \space cos \space C-a \space  \space (4)

Razonando de forma similar sobre las otras alturas del triángulo ABC se obtiene:

a^2=b^2+c^2-2bc \space cos A

b^2=a^2+c^2-2ac \space cos B

Otra altura

x

h_A

Teorema del seno

Teorema del coseno

Elige teorema:

Reiniciar construcción

E1 = (-4, 6)
E2 = (3, -0.5)
r1 = OrthogonalLine(E1, xAxis)
r2 = OrthogonalLine(E1, yAxis)
r3 = OrthogonalLine(E2, xAxis)
r4 = OrthogonalLine(E2, yAxis)
E3 = Intersect(r2, r3)
E4 = Intersect(r1, r4)
poli1, e1, e3, e2, e4 = Polygon(E1, E3, E2, E4)
A_1 = PointIn(poli1)
B_1 = PointIn(poli1)
C_1 = PointIn(poli1)
polígono1, c, a, b = Polygon(A_1, B_1, C_1)
rhA = OrthogonalLine(A_1, a)
rBC = Line(C_1, B_1)
H = Intersect(rhA, rBC)
g = Segment(H, C_1)
C = Angle(B_1, C_1, A_1)
B = Angle(C_1, B_1, A_1)
th: 0
rAB = Line(A_1, B_1)
rhC = OrthogonalLine(C_1, rAB)
D = Intersect(rAB, rhC)
h_C = Segment(D, C_1)
l = Segment(D, A_1)
texto4: "a = " + a + ""
A = Angle(C_1, A_1, B_1)
sena: sin(A)
texto5: "sen A = sen " + A + " =" + sena + ""
senb: sin(B)
texto6: "b = " + b + ""
texto7: "sen B = sen " + B + " = " + senb + ""
texto8: "c = " + c + ""
senc: sin(C)
texto9: "sen C = sen " + C + " = " + senc + ""
h: a / sena
i: b / senb
j: c / senc
texto10: "\frac{a}{sen \space A}=" + (LaTeX[h]) + ""
texto11: "\frac{b}{sen \space B}=" + (LaTeX[i]) + ""
texto12: "\frac{c}{sen \space C}=" + (LaTeX[j]) + ""
k: false
m: false
o: false
n: false
p = Line(A_1, C_1)
q = OrthogonalLine(B_1, p)
E = Intersect(p, q)
r = Segment(E, B_1)
s = Segment(H, B_1)
polígono2, b_1, a_1, h_1 = Polygon(A_1, H, B_1)
polígono3, c_1, a_2, h_2 = Polygon(A_1, H, C_1)
polígono4, a_3, c_2, d_1 = Polygon(C_1, D, A_1)
polígono5, c_3, b_2, d_2 = Polygon(B_1, D, C_1)
polígono6, e_1, b_3, a_4 = Polygon(B_1, A_1, E)
polígono7, c_4, b_4, e_2 = Polygon(B_1, E, C_1)
t = Segment(E, A_1)
F = Midpoint(A_1, B_1)
G = Midpoint(C_1, B_1)
I = Midpoint(A_1, C_1)
f_1 = Segment(H, B_1)
J = Midpoint(H, B_1)
g_1 = Segment(H, C_1)
K = Midpoint(H, C_1)
L = Midpoint(H, B_1)
M = Midpoint(A_1, H)
cosa: cos(A)
cosb: cos(B)
cosc: cos(C)
texto44: "cos A = " + cosa + ""
texto46: "cos B = " + cosb + ""
texto47: "cos C = " + cosc + ""
a2: a^(2)
texto48: "a^2 =" + (LaTeX[a2]) + ""
a2d: b^(2) + c^(2) - (((2 * b) * c) * cosa)
texto49: "b^2+c^2-2bc \space cos \space A=" + (LaTeX[a2d]) + ""
b2: b^(2)
texto50: "b^2=" + (LaTeX[b2]) + ""
b2d: a^(2) + c^(2) - (((2 * a) * c) * cosb)
c2: c^(2)
c2d: a^(2) + b^(2) - (((2 * a) * b) * cosc)
texto52: "c^2=" + (LaTeX[c2]) + ""
texto51: "a^2+c^2-2 a c  \space cos \space B=" + (LaTeX[b2d]) + ""
texto53: "a^2+b^2-2ab \space cos \space C=" + (LaTeX[c2d]) + ""
d: false
N = Midpoint(B_1, H)
O = Midpoint(b_1)