三角形的31个面积公式
三角形的31个面积公式
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C (x 3, y 3)
b
a
A (x 1, y 1) B (x 2, y 2)
令p =
1
(a +b +c ) △ABC 2
三边上的高分别为h a , h b , h c
△ABC 的外接圆半径为R ,内切圆半径为r AB, AC, BC 边上的中线分别是m c , m b , m a ∠A, ∠B, ∠C 的平分线长分别为t a , t b , t c
∠A, ∠B, ∠C 的外角平分线长分别为t ′a , t ′b , t ′
c
∠A, ∠B, ∠C 所对的旁切圆半径分别为r a , r b , r c
S △ABC
=1
2ah 11a =2bh b =ch c =√
2p (p −a )(p −b )(p −c ) =pr
=1
12 bc sin A = 2ac
x 11 sin B =1
ab sin C 2= 1 y 1 1 2
x 2y 2 x 3y 3
1
1
(1)(2)(3)(4)
(5)
S △√ABC =1 2
AC · AB 2−AC ·AB 2=a 2sin B sin C
b 2sin A sin C c 2sin A sin B ==
=2R 2sin A sin B sin C
=h 2c sin C h 2a sin A h 2b sin B 2sin A sin B =2sin B sin C =2sin A sin C =abc 4R
=a 2sin 2B +b 2sin 2A b 2sin 2C +c 2sin 2B a 2sin 2C +c 2sin 2A 4=4=
4
=(a 2−b 2
) sin A sin B (a 2−c 2) sin A sin C (b 2−c 2) sin B sin C 2sin (A −B ) =2sin (A −C ) =
2sin (B −C ) =2abc A B C a (+b +c cos 2cos 2cos 2=a 222
) √sin A +b sin B +c sin C sin A B C 2sin 2sin
2=12a 2b 2−a 2+b 2−c 222
=4Rr cos A B C
2cos 2cos
2=4Rp sin A B C
2sin 2sin
2
=p (p −a ) tan A B C
2=p (p −b ) tan 2=p (p −c ) tan
2=p 2tan A 2tan B C
2tan
2
=18(4m 2a −a 2) tan A =18(4m 2b −b 2) tan B =12
28
(4m c −c ) tan C =b 2+c √(p −b )(p −c ) t a +c √a +b √
a =2(p −a )(p −c ) t b =2(p −a )(p −b ) t c
=t a t b t c abcp
(a +b )(b +c )(c +a ) =|b −c |√p (p −a ) t ′|a −c |√|a −b |√ 2a =2p (p −b ) t ′
b =p (p −c ) t ′c
b 2−c 2 2 2
24bc t a a t ′−c 2 a =4ac t a b t ′−b 2= b =4ab
t c t ′c
=rr A B C
a cot 2=rr b cot 2=rr c cot
2
2
(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)
S =√
△ABC a b c
=r a r b r c
a b b c c a
=aR sin B sin C =bR sin A sin C =cR sin A sin B
=1(b 2+c 2−a 2) tan A =1(a 2+c 2−b 2) tan B =1(a 2
+b 2) 444−c 2tan C =r a (p −a ) =r b (p −b ) =r c (p −c ) =
a 2+b 2+c 2
4(cot A +cot B +cot C )
3
(26)(27)(28)(29)(30)(31)