The setup is a follows: You have three sound receivers (microphones) at known locations $P_1(x_1,y_1), P_2(x_2,y_2), P_3(x_3,y_3)$ that all lie on a plane. A sound source in the same plane produces a blip (a short high-pitched sound made by an electronic device) at an unknown time $t_0$. The blip is received at the three receivers at times $t_1, t_2, t_3$ respectively. Use this information to determine the location of the sound source. It is assumed that the speed of sound is known.
My attempt:
If $c$ is the speed of sound, and if $d_i$ is the distance travelled by sound from the source to the $i$-th receiving station, then
$$ d_i = c \ (t_i - t_0) = a_i - k $$
where $a_i = c \ t_i $ and is known for $i=1,2,3$ and $ k = c \ t_0 $ and is unknown.
On the other hand, we have for each of the three stations:
$d_i^2 = (P - P_i) \cdot(P - P_i) $
There are three unknowns here which are $k , P_x , P_y $
Now it is a matter of solving these three quadratic equations for the three unknowns.
These equations are relatively easy to solve because by subtracting pairs of equations we get
$ -2 P \cdot (P_1 - P_2) + P_1 \cdot P_1 - P_2 \cdot P_2 = a_1^2 - a_2^2 - 2 k (a_1 - a_2 ) \tag{1} $
and
$ -2 P \cdot (P_1 - P_3) + P_1 \cdot P_1 - P_3 \cdot P_3 = a_1^2 - a_3^2 - 2 k (a_1 - a_3 ) \tag{2}$
Equations $(1),(2)$ are linear equations of planes. Solving the system produces a line in space of the form
$ (P_x, P_y, k) = X_0 + \lambda X_1 $
where both $X_0, X_1$ are known quantities while $\lambda$ is a free parameter $\in \mathbb{R} $
To determine $\lambda$, substitue $P_x, P_y, k$ into any of the original three quadratic equations. This will give a scalar quadratic equations in $\lambda$ which produces two possible solutions.
I think this is similar to your idea but using the concept of circles.
Since we do know the distance of the source from each receiver, we can write the eqn of 3 circles such that each receiver is the centre and the distance to the source as its radius. Now finding its radial centre would be our answer.
I am not sure so please feel to let me know if I went wrong anywhere