Equation of circle circumscribing a triangle

Question

Th equation of a circle circumscribing a triangle whose sides in order are represented by the lines ${L_1} = 0$, ${L_2} = 0$, ${L_3} = 0$, is given by ${L_1}{L_2} + \lambda {L_2}{L_3} +\mu {L_3}{L_1}= 0$ provided the coefficients of ${x^2}$ and ${y^2}$ are equal and the coefficient of $xy = 0$. My question is why the equation cannot be just of form ${L_1}{L_2} + \lambda {L_2}{L_3}=0$ leaving the $\mu {L_3}{L_1}$ term because leaving that we still have in all cases of two of the $L_1 ,L_2, L_3$ to be zero both terms would go zero and satisfy the equation . Why the third term is needed ?