Animated Solution for Mathematics - Complex Numbers: If one the vertices of the square circumscribing the circle $|z-1|=\sqrt{2}$ is $2+\sqrt{3}i$. Find the other vertices of the square.

We begin by examining the given equation of the circle: $|z-1|=\sqrt{2}$. This equation represents the heartbeat of our geometry.

From this standard form, we extract two vital pieces of information. The center of the circle, which we denote as $z_0$, is $1+0i$, or simply $1$. The radius $r$ is $\sqrt{2}$.

Consider the square that circumscribes this circle. The center of the square is identical to the center of the circle, $z_0$.

The distance from the center to the midpoint of a side is the radius $r$. The distance from the center to a vertex, which we call $R$, is the hypotenuse of a right-angled triangle with legs $r$ and $r$.

By the Pythagorean theorem, the distance $R$ is:

Since our radius $r=\sqrt{2}$, the distance $R$ is:

Every vertex of this square is exactly $2$ units away from the center.

We are given the vertex $z_1=2+\sqrt{3}i$. Let us calculate the distance $|z_1-z_0|$ to verify its position:

The magnitude is:

The distance matches perfectly, confirming $z_1$ is indeed a vertex of the square.

In the complex plane, multiplying by $e^{i\theta }$ rotates a vector by an angle $\theta$. Since a square has four-fold symmetry, the vertices are separated by $90^{\circ }$, or $\frac{\pi }{2}$ radians.

To find the other vertices, we rotate the vector $(z_1-z_0)$ by $90^{\circ }$, $180^{\circ }$, and $270^{\circ }$.

For the second vertex $z_2$, we rotate by $90^{\circ }$ using the factor $e^{i\pi /2}=i$:

Thus, $z_2=1-\sqrt{3}+i$.

For the third vertex $z_3$, we rotate by $180^{\circ }$ using the factor $e^{i\pi }=-1$:

Thus, $z_3=-\sqrt{3}i$.

For the fourth vertex $z_4$, we rotate by $270^{\circ }$ using the factor $e^{i3\pi /2}=-i$:

Thus, $z_4=1+\sqrt{3}-i$.

By utilizing the inherent rotational symmetry of the complex plane, we have identified all vertices without resorting to laborious systems of equations. The vertices of the square are: