Using properties of determinants, prove the following: \begin{vmatrix} a^2 & bc & ac+c^2 \ a^2+ab & b^2 & ac \ ab & b^2+bc & c^2 \end{vmatrix} = 4a^2b^2c^2

Adding row 2 and row 3 to row 1 and taking out common factor, we get:
2\begin{vmatrix} a^2+ab+bc & ac+c^2+b^2+bc & a^2+ab+ac+c^2 \ a^2+ab & b^2 & ac \ ab & b^2+bc & c^2 \end{vmatrix}
Subtracting row 3 from row 1, we get:
2\begin{vmatrix} a^2 & ac & a^2+ac \ a^2+ab & b^2 & ac \ ab & b^2+bc & c^2 \end{vmatrix}
Subtracting row 1 from row 2, we get:
2\begin{vmatrix} a^2 & ac & a^2+ac \ ab & b^2-ac & -a^2 \ ab & b^2+bc & c^2 \end{vmatrix}
Subtracting row 2 from row 3, we get:
2\begin{vmatrix} a^2 & ac & a^2+ac \ ab & b^2-ac & -a^2 \ 0 & bc+ac & c^2+a^2 \end{vmatrix}
the above determinant on expansion gives:
2a^2(b^2c^2+a^2b^2+abc^2+a^2bc)-2ac(ab(c^2+a^2)-a^2(bc+ac))+2(a^2+ac)(ab(bc+ac)-b^2(bc-ac)) = 4a^2b^2c^2