嵌入不等式

设A＋B＋C＝(2k+1)π

x,y,z∈R

则有

x^2+y^2+z^2>=2yzcosA+2xzcosB+2xycosC

等号成立当且仅当x:y:z=sinA:sinB:sinC

嵌入不等式的等价形式(1)

设A＋B＋C＝(2k+1)π

x,y,z∈R

则有

xy[sin(C/2)]^2+zx[sin(B/2)]^2+yz[sin(A/2)]^2>=(1/4)(2xy+2yz+2zx-x^2-y^2-z^2)

等号成立当且仅当x:y:z=sinA:sinB:sinC

嵌入不等式的等价形式(2)

设A＋B＋C＝(2k+1)π

x,y,z∈R

则有

(x+y+z)^2>=4{xy[cos(C/2)]^2+zx[cos(B/2)]^2+yz[cos(A/2)]^2}

等号成立当且仅当x:y:z=sinA:sinB:sinC