I did a JavaScript interactive picture of the Malfatti circles of a triangle. The user can drag the vertices of the triangle and the Malfatti circles are updated accordingly.


Now, I would like to restrict the transformations of the original triangle in order that they preserve the outer Soddy circle of the Malfatti circles (the goal is to do an interactive Apollonian gasket with a fixed exterior circle).


In other words, let's say that the outer Soddy circle of the Malfatti circles of a triangle is the *Soddy-Malfatti circle* of this triangle; what are all the triangles which have a given Soddy-Malfatti circle ?




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EDIT
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[Here](https://laustep.github.io/stlahblog/frames/paperjs_Malfatti.html) is the interactive picture (click and drag). How to move the triangle in such a way that the outer circle remains fixed?



Triangles with a given outer Soddy circle of the Malfatti circles
At the request of the poster, I am expanding my comment to an answer. We are given a circle $\cal C$ which we take to be the unit one with centre at the origin of the coordinate plane. Now if $PQR$ is a triangle in the plane, there is a unique triangle $ABC$ with $A=(0,0)$, $B=(1,0)$ and $C=(p,q)$ to which it is directly similar. (This is a common ploy in triangle geometry which seems to have been discovered and forgotten several times but goes back at least to Euler). We compute the Soddy-Malfatti circle of $ABC$
and denote its centre by $M(p,q)$, its radius by $r(p,q)$. If we translate the original triangle by $-M(p,q)$, then dilate it by $\frac 1 {r(p,q)}$, we obtain one with $\cal C$ as Soddy-Malfatti circle. Repeating this for all values of $p$ and $q$ gives a two parameter family of triangles with the required properties.


