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Bissektrisa (qizil) Bissektrisa (lotincha : — "ikki marta va lotincha : sectio — kesish, kesuvchi) — biror burchak uchidan chiquvchi va uni teng ikkiga boʻluvchi nur bo'ladi. U berilgan burchakdan chiqib uni teng ikkiga boʻladi(60⁰=30⁰.30⁰). 3a tomondan bissektrissa o'tqazilsa ular bir joyda kesishadi.
Quyidagi formulalarni olish uchun Styuart teoremasidan foydalanishingiz mumkin.
Uchburchak ABC bissektrisasi.Quyidagi formulalar Styuart teoremasidan kelib chiqgan.
l
c
=
a
b
(
a
+
b
+
c
)
(
a
+
b
−
c
)
a
+
b
=
2
a
b
p
(
p
−
c
)
a
+
b
{\displaystyle l_{c}={{\sqrt {ab(a+b+c)(a+b-c)}} \over {a+b}}={\dfrac {2{\sqrt {abp(p-c)}}}{a+b}}}
p
{\displaystyle p}
yarim perimetr .
l
c
=
a
b
−
a
l
b
l
{\displaystyle l_{c}={\sqrt {ab-a_{l}b_{l}}}}
l
c
=
2
a
b
cos
γ
2
a
+
b
{\displaystyle l_{c}={\frac {2ab\cos {\frac {\gamma }{2}}}{a+b}}}
l
c
=
2
a
l
b
l
cos
γ
2
a
l
2
+
b
l
2
−
2
a
l
b
l
cos
(
γ
)
{\displaystyle l_{c}={\dfrac {2a_{l}b_{l}\cos {\dfrac {\gamma }{2}}}{\sqrt {a_{l}^{2}+b_{l}^{2}-2a_{l}b_{l}\cos {(\gamma })}}}}
l
c
=
h
c
cos
α
−
β
2
{\displaystyle l_{c}={\frac {h_{c}}{\cos {\frac {\alpha -\beta }{2}}}}}
Uchburchakning uchta burchak bissektrisalari uchun
A
{\displaystyle A}
B
{\displaystyle B}
C
{\displaystyle C}
l
a
,
l
b
,
{\displaystyle l_{a},l_{b},}
l
c
{\displaystyle l_{c}}
[1]
(
b
+
c
)
2
b
c
l
a
2
+
(
c
+
a
)
2
c
a
l
b
2
+
(
a
+
b
)
2
a
b
l
c
2
=
(
a
+
b
+
c
)
2
{\displaystyle {\dfrac {(b+c)^{2}}{bc}}l_{a}^{2}+{\dfrac {(c+a)^{2}}{ca}}l_{b}^{2}+{\dfrac {(a+b)^{2}}{ab}}l_{c}^{2}=(a+b+c)^{2}}
w
c
2
=
a
w
⋅
b
w
−
a
b
=
C
E
2
=
B
E
⋅
A
E
−
a
b
{\displaystyle w_{c}^{2}=a_{w}\cdot b_{w}-ab=CE^{2}=BE\cdot AE-ab}
Inmarkazi (uchburchakning uchta ichki bissektrisasining kesishish nuqtasi)
A
{\displaystyle A}
b
+
c
a
{\displaystyle {\frac {b+c}{a}}}
a
,
b
,
c
{\displaystyle a,b,c}
A
,
B
,
C
{\displaystyle A,B,C}
α
,
β
,
γ
{\displaystyle \alpha ,\beta ,\gamma }
A
,
B
,
C
{\displaystyle A,B,C}
h
c
{\displaystyle h_{c}}
c
{\displaystyle c}
uchburchak balandligi .
l
c
{\displaystyle l_{c}}
c
{\displaystyle c}
a
l
,
b
l
{\displaystyle a_{l},b_{l}}
l
c
{\displaystyle l_{c}}
c
{\displaystyle c}
w
c
{\displaystyle w_{c}}
C
{\displaystyle C}
A
B
{\displaystyle AB}
a
w
,
b
w
{\displaystyle a_{w},b_{w}}
w
c
{\displaystyle w_{c}}
c
=
A
B
{\displaystyle c=AB}
Agar median
m
{\displaystyle m}
h
{\displaystyle h}
t
{\displaystyle t}
c
h
e
g
a
r
a
l
a
n
g
a
n
{\displaystyle chegaralangan}
[2]
4
R
2
h
2
(
t
2
−
h
2
)
=
t
4
(
m
2
−
h
2
)
.
{\displaystyle 4R^{2}h^{2}(t^{2}-h^{2})=t^{4}(m^{2}-h^{2}).}
OʻzME . Birinchi jild. Toshkent, 2000-yil
↑ Simons, Stuart. Mathematical Gazette 93, March 2009, 115—116.
↑ Altshiller-Court, Nathan, College Geometry , Dover Publ., 2007.
