# 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 $$a+b+c$$.

×