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A.
Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff $\bar{z_1}z_2+\bar{z_2}z_3+\bar{z_3}z_1=r$ which $r$ is a real number.
B.
Prove three distinct complex numbers $z_1,z_2,z_3$ lie on a straight line iff there are non-zero $\lambda_1,\lambda_2,\lambda_3$ such that $\lambda_1+\lambda_2+\lambda_3=0$ and $\lambda_1z_1+\lambda_2z_2+\lambda_3z_3=0$

I already knew one condition is $z_1-z_2=r(z_2-z_3)$ for some real number . But don't know how to infer these two conditions. Any hints would be helpful.

David C. Ullrich
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Jaqen Chou
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1 Answers1

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I propose a solution that is different from the solution given in Why $\bar{z_1}z_2+\bar{z_2}z_3+\bar{z_3}z_1 \in \mathbb R \iff z_1, z_2, z_3 \text{ are along the same line}$ :

Do you know that the equation of the straight line passing through $z_1$ and $z_2$ is

$$\begin{vmatrix}z_1&z_2&z\\ \overline{z_1} & \overline{z_2}& \overline{z}\\1&1&1\end{vmatrix}=0 \ ?$$

(see for example Equation of line in form of determinant)

Thus $z=z_3$ belongs to this line iff it verifies the above equation, giving the following condition :

$$\begin{vmatrix}z_1&z_2&z_3\\ \overline{z_1} & \overline{z_2}& \overline{z_3}\\1&1&1\end{vmatrix}=0\tag{1}$$

which, once developed, is equivalent to

$$\overline{z_2}z_1+\overline{z_1}z_3+\overline{z_3}z_2=\overline{z_1}z_2+\overline{z_3}z_1+\overline{z_2}z_3 \tag{2}$$

But the LHS of (2) is the conjugate of its RHS. Thus this is possible iff this RHS is real, finding the given condition.

Jean Marie
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