已知三角形三边长为a,b,c,面积为S,求证：a^2+b^2+c^2>=4根号3S.

Question

已知三角形三边长为a,b,c,面积为S,求证：a^2+b^2+c^2>=4根号3S.

Answer 1

因为a^2=b^2+c^2-2bccosA
S=(1/2)bcsinA
则a^2+b^2+c^2-4√3S
=b^2+c^2-2bccosA+b^2+c^2-4√3*(1/2)bcsinA
=2b^2+2c^2-2bccosA-2√3bcsinA
=2b^2+2c^2-4bc[(1/2)cosA+(√3/2)sinA]
=2b^2+2c^2-4bc+4bc-4bccos(60-A)
=2(b-c)^2+4bc[1-cos(60-A)]
-120 < 60-A < 60
-1/2 < cos(60-A) ≤ 1
0 ≤ 1-cos(60-A) < 3/2
所以
a^2+b^2+c^2-4√3S = 2(b-c)^2+4bc[1-cos(60-A)] ≥0
当b=c且A=60时,即等边三角形时,等号成立