Prove that `|{:(1,,ab,,(1)/(a)+(1)/(b)),(1 ,,bc,,(1)/(b)+(1)/(c)),( 1,, ca,,(1)/(c)+(1)/(a)):}|=0`

Explanation

To prove that the determinant

\[
D = \begin{vmatrix}
1 & ab & \frac{1}{a} + \frac{1}{b} \\
1 & bc & \frac{1}{b} + \frac{1}{c} \\
1 & ca & \frac{1}{c} + \frac{1}{a}
\end{vmatrix}
...