{Bx,By,Bz} → {Stlx,Stly,Stlz}, {-Bx,By,Bz} → {Strx,Stry,Strz}

Solve[{(x - Rx)/(Bx - Rx) == (y - Ry)/(By - Ry) == (z - Rz)/(Bz - Rz), Tx x + Ty y + Tz z == Tx Cx + Ty Cy + Tz Cz}, {x, y, z}]