6. 已知 $\triangle ABC$ 是等腰三角形,过 $\triangle ABC$ 的一个顶点的一条直线,把 $\triangle ABC$ 分成两个小三角形,如果这两个小三角形也是等腰三角形,试求出 $\triangle ABC$ 各内角的度数. (写出所有的情况)
答案：解:一共有4种情况.
如答图①,$\triangle ABC$是等腰三角形,$AB = AC$,线段$AD$过顶点$A$.根据题意,知$\triangle ABD,\triangle ACD$是等腰三角形,且$AD = BD$,$AD = CD$,那么$\angle B=\angle BAD=\angle CAD=\angle C$,利用三角形内角和定理,可知$\angle B+\angle BAD+\angle CAD+\angle C = 180^{\circ}$,解得$\angle B=\angle BAD=\angle CAD=\angle C = 45^{\circ}$,$\angle BAC = 90^{\circ}$.
如答图②,$AB = AC = CD$,$AD = BD$,设$\angle B = x$,则$\angle BAD=\angle ACB = x$,$\therefore\angle ADC = 2x$,$\because AC = CD$,$\therefore\angle CAD = 2x$,$\therefore\angle BAC = 3x$,$\therefore x + 3x + x = 180^{\circ}$,解得$x = 36^{\circ}$,$\therefore\angle B=\angle C = 36^{\circ}$,$\angle BAC = 108^{\circ}$.
如答图③,$AB = AC$,$AD = BD = BC$,设$\angle A = x$,则$\angle ABD = x$,$\therefore\angle BDC = 2x$,$\because BD = BC$,$AB = AC$,$\therefore\angle ABC=\angle ACB = 2x$,$\therefore x + 2x + 2x = 180^{\circ}$,$\therefore x = 36^{\circ}$,$\therefore\angle A = 36^{\circ}$,$\angle ABC=\angle C = 72^{\circ}$.
如答图④,$AB = AC$,$BC = CD$,$AD = BD$.设$\angle A = x$,则$\angle ABD = x$,$\angle BDC=\angle DBC = 2x$,$\therefore\angle ABC=\angle C = 3x$,$\therefore x + 3x + 3x = 180^{\circ}$,$\therefore x=\frac{180^{\circ}}{7}$,$\therefore\angle A=\frac{180^{\circ}}{7}$,$\angle ABC=\angle C=\frac{540^{\circ}}{7}$.
