Question

(本小题满分14分)已知⊙的半径是，它的内接三角形中，有成立，求角的大小及三角形面积的最大值.

Answer 1

C=，（S_△ABC）_max =

已知等式两边同乘以2R，则有a²－c²=ab－b²，                         2分
即a²＋b²－c²=ab，                                                                              3分
∴cosC=.
∴C=，A＋B=.                                                                                    6分
∴S_△ABC=absinC=ab=R²sinA·sinB                                                       8分
= R²sinA·sin（－A）
= R²sinA(cosA + sinA)                                     9分
= R²sin²A + R²sinA cosA
=                                         10分
=
=                                          11分
 ≤.              12分
即（S_△ABC）_max =.                                                        　     14分