记 △ABC的内角A,B,C的对边分别为a.,b.,c,已知b 2 =ac,点D在边AC 上,BDsin∠ABC = asinC.
(1)证明:BD = b:
(2)若AD = 2DC .求cos∠ABC.
(1) 由正弦定理
得 ,即 =
又由 BD =asinc,得BD =asinc,
即 BD=b
(2) 由 AD=2DC,将 =2 ,即 = =
| | 2 | | 2 + | | 2 +
= c 2 + a 2 + c a
-11ac+3 =0
a= c或a= c
① cos =
综上
cos =