Let's say you have the three points (x1,y1), (x2,y2) and (x3,y3) as reference and (xa,ya) unknown. The unknown are two: xa and ya then you need two linearly independent equations
The distance from A to 1,2 and 3 are:
d1=sqrt[(xa-x1)^2+(ya-y1)^2]
d2=sqrt[(xa-x2)^2+(ya-y2)^2]
d3=sqrt[(xa-x3)^2+(ya-y3)^2]
So we have three differences:
D12 = d2-d1
D13 = d3-d1
D32 = d2-d3
each pair of them are linearly independents, while the third is the linear combination of the other two. We can choose arbitrary the first and the second, then:
D12 = sqrt[(xa-x2)^2+(ya-y2)^2]-sqrt[(xa-x1)^2+(ya-y1)^2]
D13 = sqrt[(xa-x3)^2+(ya-y3)^2]-sqrt[(xa-x1)^2+(ya-y1)^2]
You can solve it using a solver of non-linear system of equations.