, . ABC.
, , , 2R. 2R = a / sin α, 2R = BC / sin A.
BD . BCD , ( , ).
, , , , , CDB CAB ( A D BC), π - CAB ( ).
. sin(π − α) = sin α, .
2R = a / sin α, 2R = BC / sin A. sin A .
2R = BC / sin A
2R = BC / (BC / DB)
2R = DB
, , DB , .
, :
.
. ( ). , - . " " sqrt(), sqrt - , .
:
, , :
a / sin α = b / sin β = c / sin γ = 2R
R -
- . " ".
XYZ =30 Z=15. YQ ZY Z XQ QZ. XY, QZ=1.5
.
XYQ ZYQ.
.
QZ / sin(QYZ) = QY / sin(QZY)
QZY = 15 , , QYZ = 180 - 90 - 15 = 75
:
- 15 sin(15) =
- 75 sin(75) =
| ( " ") QZ / sin(75) = QY / sin(15) QZ / ((√3 + 1) / (2√2)) = QY / ((√3 - 1) / (2√2)) QZ * 2√2 / (√3 + 1) = QY * 2√2 / (√3 - 1) 3√2 (√3 - 1) = QY * 2√2 (√3 + 1) QY = 3√2 (√3 - 1) / (2√2 (√3 + 1)) QY = 3/2 (√3 - 1) / (√3 + 1) |
, XY .
|
|
|
QY / sin(30) = XY / sin(90)
:
- 30 sin(30) = 1 / 2
- 90 sin(90) = 1
QY = XY sin (30)
3/2 (√3 - 1) / (√3 + 1) = 1/2 XY
XY = 3 (√3 - 1) / (√3 + 1) ≈ 0.8
: 0,8 3 (√3 - 1) / (√3 + 1)
( 2)
. ( ). , - .
. " ".
ABC 16. 30 . 105 . .
.
, :
a / sin α = b / sin β = c / sin γ
BC / sin α = AB / sin γ
, , 180 .
= 180 - 30 -105 = 45 .
:
BC / sin 30 = 16 / sin 45
BC = 16 sin 30 / sin 45
, :
BC = (16 * 1 / 2) / √2/2 = 16 / √2 ≈ 11,3
: 16 / √2
.
ABC = α, = β, = 7, - .
.
. , :
BC / sin α = AB / sin β
7 / sin α = AB / sin β
AB = 7 sin β / sin α
ABH. BH - , . AHB = 90 .
ABH = 180 - 90 - α. = 90 - α.
:
AB / sin 90 = AH / sin (90 - α)
sin 90 = 1,
AB = AH / sin (90 - α)
sin(90 - α) = cos α,
AB = AH / cos α
AB
7 sin β / sin α = AH / cos α
AH sin α = 7 sin β cos α
AH = 7 sin β cos α / sin α
, cos α / sin α = ctg α,
AH = 7 sin β ctg α
: 7 sin β ctg α
