Prove that 4 points belong to the same circle by using complex numbers

Question

I have Z1, Z2, Z3, Z4 and they are all complex numbers. I want to prove that they belong on the same circle(C) and its center is O where O = 3

How do I do that? (They actually have equations, I just don't want to write them here because they don't matter, what matters is the way)

The exercise solved by calculating the length between Z1 and O, then Z2 and O, then Z3 and O, then Z4 and O. Then they all gave the same result which means they all belong on the same circle.

This is what I don't get it. I guess this would work if we said that a point of these points belongs to the same circle. But the thing is that we have to prove them all.

What tells me that they are not inside or outside the circle?

Answer 1

I don't understand your doubt. Anyway, if the distance for each $Z_k$ to $O$ (with $k\in\{1,2,3,4\}$) is equal to some number $\rho$, the each $Z_k$ belongs to the circle centered at $O$ with radius $\rho$.

Answer 2

A circle passing through three points $z_1,z_2,z_3$ in the complex plane must have the formula

$az\overline z+bz+c\overline z+d=0$

From this we obtain the determinantal formula $$\begin{vmatrix}z\overline z & z & \overline z & 1 \\ z_1\overline z_1 & z_1 & \overline z_1 & 1 \\ z_2\overline z_2 & z_2 & \overline z_2 & 1 \\ z_3\overline z_3 & z_3 & \overline z_3 & 1\end{vmatrix}=0$$ All we need to do is put $z_4$ for the $z$ variable in the top row and see if the above relation holds at that point: $$\begin{vmatrix}z_4\overline z_4 & z_4 & \overline z_4 & 1 \\ z_1\overline z_1 & z_1 & \overline z_1 & 1 \\ z_2\overline z_2 & z_2 & \overline z_2 & 1 \\ z_3\overline z_3 & z_3 & \overline z_3 & 1\end{vmatrix}=0.$$

If this determinant is expanded by minors, differences between complex conjugates emerge and therefore the determinant will be purely imaginary. It is therefore sufficient to track the imaginary part.

Answer 3

This statement:

The four points $z_1, z_2, z_3, z_4$ all belong to the same circle $C.$

... is equivalent to this:

Pick any one of the four points $z_1, z_2, z_3, z_4.$ Let $C$ be the circle about $O$ that passes through your chosen point. The other three points also belong to the circle $C.$

You don't have to worry that all four points are inside the circle $C,$ or that they are all outside, because of all the circles with center $O,$ circle $C$ is the one that passes through all four points.

Note that this is not a trivial statement. Given a point $O$ and four arbitrary points, you might have three points at distance $r$ from $O,$ therefore all three are on the same circle about $O$, but the fourth point might be inside or outside that circle. Or the four points could be at four different distances from $O$ so that there is no circle about $O$ to which even two of the points belong.