Multiply:
$(a b + b c + c a)$ by $(a b - b c - c a)$

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Solution

$(a b + b c + c a) (a b - b c - c a)$
$= [a b + (b c + c a)] [a b - (b c + c a)]$
$= (a b)^{2} - (b c + c a)^{2}$
$= a^{2} b^{2} - (b^{2} c^{2} + c^{2} a^{2} + 2 a b c^{2})$
$= a^{2} b^{2} - b^{2} c^{2} - c^{2} a^{2} - 2 a b c^{2}$

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∣ ∣ ∣ ∣ \begin{matrix} - b c & c a + a b & c a + a b a b + b c & - c a & a b + b c b c + c a & b c + c a & - a b \end{matrix} ∣ ∣ ∣ ∣ = (\sum a b)^{3}

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If D is any point on the side BC of $Δ A B C,$ then

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