107全國高中聯招

複選題9.
若\(\displaystyle \frac{b+c}{2a}=\frac{a-c}{2b}=\frac{a-b}{2c}\)，求\(\displaystyle \frac{a}{a+b+c}\)的值？(A)1　(B)\(-1\)　(C)\(\displaystyle \frac{1}{2}\)　(D)\(\displaystyle -\frac{1}{2}\)
[解答]
給老師您參考(多選九)
\(\displaystyle \frac{b+c}{2a}=\frac{a-c}{2b}=\frac{a-b}{2c}=\frac{2a}{2a+2b+2c}=\frac{a}{a+b+c}=t\)
\(\cases{b+c=2at\cr a-c=2bt \cr a-b=2ct}\)
\(\cases{(-2t)a+b+c=0\cr a+(-2t)b-c=0 \cr a-b-(2t)c=0}\)
除了\((0,0,0)\)以外還有其他解(分母\(a,b,c\ne 0\))
\(\Delta =\Bigg|\; \matrix{-2t&1&1 \cr 1&-2t&-1 \cr 1 &-1&-2t}\Bigg|\;=0\)
\((t+1)(8t^2-8t+2)=0\)
\(\displaystyle t=-1\)或\(\displaystyle \frac{1}{2}\)