Using distance formula or otherwise, prove that the point $- 2 + 3 i, - 1 + 2 i$ and $5 - 4 i$ are collinear.

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Solution

Let $A \equiv (- 2 + 3 i), B \equiv (- 1 + 2 i), C \equiv (5 - 4 i)$

Distance between A and B = $| - 1 + i | = \sqrt{2}$

Distance between B and C = $| - 6 + 6 i | = \sqrt[6]{2}$

Distance between A and C = $| - 7 + 7 i | = \sqrt[7]{2}$

$∴ A B + B C = A C$

Hence, they are collinear.

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