\overrightarrow{DA_1}=\overrightarrow{DA}+\overrightarrow{AA_1}=\overrightarrow{DA}+k\overrightarrow{DD_1}=\frac{a+b+c+ka}{a+b+c}\overrightarrow{DA}+\frac{kb}{a+b+c}\overrightarrow{DB}+\frac{kc}{a+b+c}\overrightarrow{DC}