How to find the third coordinate of a triangle using known two coordinates and distance to 3rd point?

Question

$A(a_1,a_2)$ is the first point, $B(b_1,b_2)$ is the second point and $d_1=BC$, $d_2=CA$ and $d_3=AB$ are known distances from each points to the other. How to find $C(c_1,c_2)$?

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Answer 1

Build the equation of circle 1 (centered at A, radius $= d_2$). It is $(x - a_1)^2 + (y - a_2)^2 = (d_2)^2$.

Do the same thing to circle 2 ((centered at B, radius $= d_3$).

Combine the two equations to get one linear equation in x and y.

Solve the linear equation and one of the circle equations.

Note: A quadratic equation has two roots.