Amazing coordinate geometry

Geometry Level 4

\[\left(\dfrac{15}{AB}\right)^2+\left(\dfrac{10}{AC}\right)^2=\left(\dfrac{6}{AD}\right)^2\]

If \(A\) is \((-5,-4)\), \(B\) is located on line \(x+3y+2=0\), \(C\) is located on line \(2x+y+4=0\) and \(D\) is located on line \(x-y-5=0\), then find the equation of line on which \(A,B,C\) and \(D\) lie and the above equation is true.

It is in the form \(ax+by+c=0\), where \(a,b,c\) are integers such that \(\gcd(a,b,c) = 1\) and \(a>0 \).

Submit your answer as \(a+b+c\).

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