AC2 = AD· AB, BC2 = BD · AB, só‹) AC2+ BC2 = (AD + BD

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àÇ 1, Tòiúi$ ABC éi,íò CD, †4ABC, 4ACD, 4CBD ·u˛

¤óNí
µó;WóNúi$2ú@iÅ5ªó (C6òQ‚ঠìÜ),  AC² = AD· AB, BC² = BD · AB, só‹) AC²+ BC² = (AD + BD) · AB = AB²

Ç 1 Ç 2

¥øj¶ªJRƒéúi$, BbJ@iúi$ÑW„pìýìÜ (iúi$ªJé NTÜ)

àÇ 2, Ê 4ABC 2, ¬ C õT(¨ CD, CE > AB k D, E, U^∠ACD =^∠B,

∠BCE = ^∠A
éÍ4ACD ∼ 4ABC ∼ 4CBE, ^∴ AC² = AD · AB, BC² =

84

y}H«ìÜDìýìÜí„p 85

BE· AB, ^AC_AB = ^{C D}_BC
7∠CDE =^∠CED =^∠A+^∠B, âìýì2ø,

cos(A + B) = cos^∠CDE =

1 2DE

CD .

∴DE = 2CD · cos(A + B) = 2^ACAB^·BCcos(A + B)

∴ AC²+ BC² = (AD + BE) · AB = (AB − DE) · AB = AB² − DE · AB = AB²−2^AC_AB^·BC cos(A+B)·AB = AB²−2AC ·BC cos(A+B) = AB²+2AC ·BC cos C

(¥êíi C N ^∠ACB
)

à¤Z„p7ìýìÜ
ÊÇ 22, JD,E½¯, † ^∠ACB Ñòi, ìýì܉ÑH«ì Ü

2. p,j¶„pìýìÜ

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Fzs_r

íòiúi$:AàÇ 3 FýíÇ$, àsj¶lG$Þ , SG$ = ¹₂(a + b)(a + b) = 2 ·¹₂ab+ ¹₂c², “cÜ) a²+ b² = c²

Ç3

‚à¥øj¶Bb.?òQ„pìýìÜ, OªJ„p sin(A + B), cos(A + B) t:

às_éiÅÌÑ 1 íòiúi$:AàÇ 4 FýíÇ$, àsj¶VlG$Þ , SG$ = ¹₂(sin A+sin B)(cos A+cos B) = ¹₂sin A cos A+¹₂sin B cos B+¹₂·1·1·sin(A+B),

“cÜ) sin(A + B) = sin A cos B + cos A sin B

86 bçfÈ 29 » 3 ‚ ¬ 94  9 ~

Ç 4 Ç 5

‚àH«ìܪø, vòiG$8ÅÑ ^q(cos A + cos B)²+ (sin A − sin B)², ‚àú i$ìýìܪ)

cos(A + B) = − cos[π − (A + B)] = 1²+ 1²− [(cos A + cos B)²+ (sin A − sin B)²]

2 · 1 · 1 ,

“cÜ) cos(A + B) = cos A cos B − sin A sin B

p,j¶ÖÍ.?òQ„pìýìÜ, O§ sin(A + B), cos(A + B)t„p¬˙

íóê, Bbzs_éi}Ñ a,b íòiúi$àÇ5[0, ‚àH«ìÜ):

c²= (a cos A + b cos B)² + (a sin A − b sin B)²

= a²cos²A+ 2ab cos A cos B + b²cos²B+ a²sin²A− 2ab sin A sin B + b²sin²B

= a² + b²+ 2ab(cos A cos B − sin A sin B)

= a² + b²+ 2ab cos(A + B) = a²+ b²− 2ab cos C.

.Øõ|, ¥wõu,H cos(A + B) t„píL¬˙
FJBbÛlàwFj¶ú

“cos A cos B − sin A sin B = cos(A + B)”tªW„p, yR|ìýìÜ, .Íÿ¸p7

=„

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