Bissektrisa
Bissektrisa (lotincha: "bi-ikki marta" — "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.
Uchburchakda bissektrisalarning uzunligi
Quyidagi formulalarni olish uchun
Styuart teoremasidan foydalanishingiz mumkin.
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}}}
, bu yerda
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}
va
C
{\displaystyle C}
uzunliklari mos ravishda
l
a
,
l
b
,
{\displaystyle l_{a},l_{b},}
va
l
c
{\displaystyle l_{c}}
, formula oʻrinli boʻla oladi.
(
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}
burchakning ichki bissektrisasini
b
+
c
a
{\displaystyle {\frac {b+c}{a}}}
ga nisbatan ajratadi,
Bu yerda:
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}).}
Adabiyotlar
uz.wikipedia.org