Moving circles, passing through 3 points I would like to know if there is an unique solution or at least a good approximation for the following problem:
I have three points in a space forming a triangle. There is nothing special for this triangle, it is not equilateral, but if we find a solution for an equilateral triangle, we can approximate it to that.
The points are (0,18.27), (24.58,34.37) and (24.82,0) and I have the exactly time when a moving circle touches each point for the first time.
The assumptions are:

*

*The circle moves at a constant speed at a constant direction;

*I only know the first time it touch the points, not the time it touch with the other half;

*The radius is, at least 20 (the circle that circumscribe the points) and at most 200;

*It can come from any direction;

The main objective:
To have some characteristic length scale of those moving circles, e.g. its radius.
I am looking for a general solution for that. The thing is: I have a spatial data from three points and I could identify a particular thing happening at in the time series from the three points. From the theory behind this natural phenomena, I could approximate it to a circle. So, I am assuming that this is a moving circle that pass through three points.
I added a figure to illustrate the problem. T_pi is the time the circle touches each point.

I also made another figure to ilustrate what I think are the variables of the problem: radius, angle, offset and speed. I am assuming that they are constant for each circle.

Feel free to ask for more information about the problem and if we can make any other assumption about that.
 A: This isn't an answer yet, just an attempt to formulate the problem.  It really should be a comment, but it won't fit in a comment box.
The center of the circle moves along a straight line with constant velocity.  We can say that at time $t$ it is at the point$$(x_0+at, y_0+bt)$$ where $x_0,y_0,a,\text{ and }b$ are unknown constants.  At times $t_1<t_2<t_3$ the point is a some fixed, but unknown distance $r$ from the points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, respectively, where $t_i,x_i,y_i$ are given for $i=1,2,3$.
That gives us $3$ equations in $5$ unknowns:
$$(x_0+at_i-x_i)^2+(y_0+bt_i-y_i)^2=r^2,\ i=1,2,3$$
I've been trying to eliminate one or more unknowns.  Of course, we can eliminate $r$, by just saying the the distances are equal, but this eliminates an equation, so we have $2$ equations in $4$ unknowns.  I don't see how to eliminate any other variables.  $(x_0, y_0)$ is the position of the moving point at time $t=0$.  We can reset the origin of the time scale, say by setting $t_i'=t_i-t_1$, but that doesn't eliminate the unknowns $x_0, y0$.  Similarly, it's only the ratio of $b$ to $a$ that's really fixed, but $\sqrt{a^2+b^2}$ is the distance the point moves in one unit of time, and I don't see any way around this.
Since we have more unknowns that equations, I tend to believe that there is not a unique solution to the system.  If the OP would gives us the values of $t_1,t_2,t_3$, I think I could demonstrate this.
EDIT
I recognize that I haven't taken cognizance of the requirements that the times are the first time the circle passes through the points, or the restrictions on the radius, but I don't think that this will change matters.  This is just a gut feeling.  I think there are infinitely many solutions to the system, and we'll be able to pick out more than one that meet the requirements.  (Of course, I'm wrong a lot.)
