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So there is a square of 50cm length. There are 3 circle's whose centre's are any 3 vertices of the square.The radius(r1,r2,r3) of these circles can be assumed i.e it is known. Now a 4th circle exists which lies on the 4th vertices and is touching the rest 3 circles tangentially.

The centre of this 4th circle has to be found(x,y). enter image description here

So i am making a rifle shooting target detection. So when a bullet hits the board there will be a mic/sensors on each corner of the target board that will capture the sound and the difference between each sensors will be directly proportional to the distance. So that 4th circle's centre is where the bullet hits, a sound wave propogates and reaches any 1 sensor first (in our case the the 4th vertice sensor),lets call this time t=0 and then to the rest of the sensors at t1=0.5sec, t2=1sec, t3=1.2sec.As we know the speed of the sound, distance can be calculated thus we know r1, r2, r3. Now i cant understand how can i find the 4th circle's centre i.e where the bullet hit. I am refering to https://github.com/ten-point-nine/freETarget/blob/master/Documentation/Theory%20of%20Operation.pdf

Rachit Juthani
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    In order to touch all these 3 circles and go through fourth vertice, there must exist 3 numbers $x,y,R$ satisfying 4 equations $d(x,y,1)=R+r_1$, $d(x,y,2)=R+r_2$, $d(x,y,3)=R+r_3$, $d(x,y,4)=R$, where $d(x,y,i)$ is distance from point $(x,y)$ to $i$-th vertice of square. 3 variables, 4 equations. In order solution to exist numbers $a$ (square side length), $r_1$, $r_2$, $r_3$ must not be arbitrary but satisfy some condition. – Ivan Kaznacheyeu Sep 07 '23 at 09:26
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    You can find the circle tangent to the other three (this is called "Problem of Apollonius") but it is not sure it will pass through the fourth vertex. – Intelligenti pauca Sep 07 '23 at 19:37
  • "4th circle's centre" = correct use of an apostrophe. "3 circle's whose centre's are" = two misuses of an apostrophe. To pluralize a regular noun, you just add an "s", No apostrophe. The apostrophe is used for contraction (e.g., writing "the circle's larger than ..." instead of "the circle is larger than") or to indicate possession ("the 4th circle's centre" instead of "the centre of the 4th circle"). – Paul Sinclair Sep 08 '23 at 21:55

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