If a, b, c are the sides of a triangle ABC and p is the perimeter of the triangle, then what is \begin{vmatrix}p+c&a&b\\ c&p+a&b\\ c&a&p+b\end{vmatrix} equal to?

Explanation

Apply the column operation C_1 \to C_1 + C_2 + C_3. The first column becomes p+a+b+c in each row. Since p = a+b+c, this is 2p. Factoring out 2p from C_1 gives 2p \begin{vmatrix}1&a&b\\ 1&p+a&b\\ 1&a&p+b\end{vmatrix}. Applying R_2 \to R_2 - R_1 and R_3 \to R_3 - R_1 transforms it into an upper triangular determinant \begin{vmatrix}1&a&b\\ 0&p&0\\ 0&0&p\end{vmatrix}, which evaluates to p^2. The result is 2p(p^2) = 2p^3.

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