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I have been assigned a problem with two point vortices:

Find two point vortices whose locations in the 2D plane and strengths $gamma_1,gamma_2$ are such that their positions remain fixed in time.

My actual Question is how to do this problem with 3 point vortices of strengths $gamma_1,gamma_2,gamma_3$ whose positions remain fixed in time.

My proof for 2 vortices. (DISCLAIMER I AM NOT SURE IF THIS IS CORRECT.)

Denote the complex potential of the two point vortices as $z_1,z_2$.
The velocity induced on $z_2$ by the vortex at $z_1$ is,
$$w(z_1)=u-iv=frac{gamma_1}{2pi i(z_1-z_2)}. $$

The velocity induced on $z_2$ by the vortex at $z_1$ is,
$$w(z_2)=u-iv=frac{gamma_2}{2pi i(z_2-z_1)}. $$

Since inviscid vorticity is simply advected by the flow, the velocity is also with which the vortex moves:

$$frac{dz_1^*}{dt}=frac{1}{2pi i}frac{gamma_1}{(z_1-z_2)}$$

$$frac{dz_2^*}{dt}=frac{1}{2pi i}frac{gamma_2}{(z_2-z_1)}$$

Where the * denotes complex conjugations.

Adding and subtracting these two equations from each other, and got

$$frac{d}{dt}(z_1+z_2)=frac{1}{2pi i}frac{gamma_1-gamma_2}{z_1-z_2} implies |z_1^*-z_2^*|=text{ constant} =2R $$

and

$$frac{d}{dt}(z_1^*-z_2^*)=frac{1}{2pi i}frac{gamma_1-gamma_2}{(z_1-z_2)} implies |z_1-z_2|=text{ constant} =2R$$

This first result shows that the center of the two vortices is staying put while the second proves that the distance does not change. Writing then

$$z_1=Rexp(itheta), text{ and} z_2=-z_1$$

any of the two above equations of motion gives

$$frac{dtheta}{dt}=frac{gamma}{2pi R^2} $$

The first result shows that the centroid of the two vortices $color{red}{text{ is fixed within a distance } (??)}$, while the second proves
that their distance does not change. Thats the best description I've got.

**Any and all help/tips proofs would be very much appreciated. Thank you! **

I believe your proof of the 2 vortex is wrong, since you have not shown that the velocities are zero.

To solve the 3 vortex case lets assume that they are originally at $z_1,z_2,z_3$ and I will use a slightly different notation than you and call $w(z_j)$ the velocity of the vortex that is at $z_j$
$$0=w(z_1) = {gamma_2 over (z_1 -z_2)} +{gamma_3 over (z_1 -z_3)}$$
$$0=w(z_2) = {gamma_1 over (z_2 -z_1)} +{gamma_3 over (z_2 -z_3)}$$
$$0=w(z_3) = {gamma_2 over (z_3 -z_2)} +{gamma_1 over (z_3 -z_1)}$$
I am ignoring the $2 i pi $ factors since the won't enter in the calculation
lets call $a_{ij} = {1 over z_i -z_j}$ and write
$$ left[begin{array}{ccc}
0& a_{12}& a_{13}
\-a_{12}& 0& a_{23}
\=a_{13}&-a_{23}&0
end{array}right]left[ begin{array}{c}gamma_1\gamma_2\gamma_3end{array}right]=left[ begin{array}{c}0\0\0end{array}right]$$
before I continue, notice that for the two vortice case the equations are

$$ left[begin{array}{cc}
0& a_{12}
\-a_{12}& 0
end{array}right]left[ begin{array}{c}gamma_1\gamma_2end{array}right]=left[ begin{array}{c}0\0end{array}right]$$ which can only be solved with zero vortex intensities.

in order to solve we can consider an easy case, when all vortices are on the $x$ axis so that $a$'s are real. Then we have a cross product $[-a_{23},a_{13},-a_{12}]times[gamma_1,gamma_2,gamma_3]=[0,0,0]$ so that by setting the $vec{gamma}$ vector to be parallel to $[-a_{23},a_{13},-a_{12}]$ we are done:
$$[gamma_1,gamma_2,gamma_3]=K[-a_{23},a_{13},-a_{12}]$$ for any real K.
Example:
$$z_1,z_2,z_3=1,2,3$$,
$$a_{23}={1over (2-3)} =-1 ; a_{13}={1over (1-3)} =-1/2; a_{12}={1over (1-2)} =-1 $$
$$[gamma_1,gamma_2,gamma_3]=K[1,-1/2,1]$$
Any 3 co-linear vortices should work (you remove the phase by adjusting the $K$), but I am not sure you can find a non co-linear solution with 3 vortices.