Bài tốn tam giác ABC có BC=a, CA=b, AB=c B1> C/m: a^2+b^2+R^2>=c^2 a^2+b^2-c^2=2abcosC 2abcosC>=-R^2 8R^2sinAsinBcosC>=-R^2 sinAsinBcosC>=-1/8 sinAsinBcos(-cos(A+B))>=-1/8 sinAsinBcos(A+B) C/m: 3sqrt(3)R>=2S, 3R ≥ S 3sqrt(3)R>=2*abc/(4R)= abc/(2R) 3sqrt(3)*2R^2>=abc = 8R^3sinAsinBsinC sinAsinBsinC RsinAsinBsinC R= abc/4S = abc/(4R^2sqrt()) 4R^3=(8R^3sinAsinBsinC)/sqrt() ½= sinAsinBsinC/sqrt() 4S=sqrt((a+b+c)(a+b-c)(b+c-a)(c+a-b))=4R^2sqrt((sinA+sinB+sinC)(sinA+sinB-sinC) (sinB+sinC-sinA) (sinC+sinA-sinB)) sinAsinBsinC ( sinA + sinB + sinC ) ( sinA + sinB − sinC ) ( sinB + sinC − sinA ) ( sinC + sinA − sinB ) = **> R= abc abc abc R sin A sin B sin C = ≥ = = 4S a+b+c R(sin A + sin B + sin C ) (a + b + c )(a + b − c)(b + c − a )(c + a − b) = 2R sin A sin B sin C sin A + sin B + sin C −− > sin A sin B sin C sin A sin B sin C ≤ −− > ≤ sin A + sin B + sin C sin A + sin B + sin C Ta có: abc ≥ (a + b − c)(b + c − a )(c + a − b) ***> sin A sin B sin C sin A sin B sin C sin A sin B sin C = = = sin A + sin B + sin C 2sin A + B cos A − B + 2sin A + B cos A + B cos C cos A − B + cos C sin C 2 2 2 2 C sin A sin B sin ≤ − − > cos A − B + sin C ≥ 4sin A sin B sin C = A− B C 2 cos + sin 2 A− B B −C C−A C A B − > cos + cos + cos + sin + sin + sin 2 2 2 C A B ≥ sin A sin B sin + sinBsinCsin + sinCsinA sin ÷ 2 2 A− B B −C C−A C A B −− > cos + cos + cos ≥ sin A sin B sin + sinBsinCsin + sinCsinA sin ÷− 2 2 2 ****> Theo hệ thức Euler d = R − Rr abc S 4S R ≥ 2r − − > ≥ = 4S p a+b+c −− > abc( a + b + c ) ≥ 16S2 = (a + b+ c)(a + b− c)(b+ c− a)(c+ a − b) −− > abc ≥ (a + b− c)(b + c− a)(c+ a − b) *****> R (abc ) 4(abc ) = = = S 16S (a + b + c)(a + b − c)(b + c − a )(c+ a − b) (a + b + c )(a + b − c)(b + c − a )(c+ a − b) 256 R (sin A sin B sin C )(sin A sin B sin C ) = 64 R (sin A + sin B + sin C )(sin A + sin B − sin C )(sinB+ sinC − sinA)(sinC+ sinA − sinB) T T = (sin A + sin B + sin C )(sin A + sin B − sin C )(sinB+ sinC− sinA)(sinC+ sinA − sinB) R2 2sin A sin B sin C = = S (sin A + sin B + sin C )(sin A + sin B − sin C )(sinB+ sinC− sinA)(sinC+ sinA − sinB) 2sin A sin B sin C = = A B C A B C B C A C A B cos cos cos 4sin sin cos 4sin sin cos 4sin sin cos 2 2 2 2 2 2 2sin A sin B sin C sin A sin B sin C = = = 2 2(sin A sin B sin C ) 2sin A sin B sin C A B C A B C 256 sin sin cos ÷ cos cos cos ÷ 2 2 2 2 Mà sin A sin B sin C ≤ 3 R2 −−> ≥ S 3