Solve for b and d in the following equation.
A triangle with sides $(a, a, b)$ has the same area and the same perimeter as a triangle with sides $(c, c, d)$ where $a, b, c$ and $d$ are positive integers and with
$$ \frac{b^2+bd+d^2}{b+d} = 7^6 $$
I have tried several basic solutions like completing and other various solutions.
Source is here
A start: If I see a quadratic and don't have a better idea, I feed it to the quadratic equation. In this case, we have $$b^2+bd+d^2=7^6(b+d)\\b^2+b(d-7^6)+d^2-7^6d=0$$ for which Alpha reports $39$ integer solutions. It only showed $5$ to start, and clicking on More Solutions didn't help. Clicking on the copiable plantext got a Mathematica output that had them all. They are $$b=-37758\quad d=64666\\ b=-37758\quad d=90741\\ b=-36015\quad d=57624\\ b=-36015\quad d=96040\\ b=-33614\quad d=50421\\ b=-33614\quad d=100842\\ b=-30576\quad d=43120\\ b=-30576\quad d=105105\\ b=-11951\quad d=13320\\ b=-11951\quad d=116280\\ b=-6174\quad d=6517\\ b=-6174\quad d=117306\\ b=0\quad d=0\\ b=0\quad d=117649\\ b=6517\quad d=-6174\\ b=6517\quad d=117306\\ b=13320\quad d=-11951\\ b=13320\quad d=116280\\ b=43120\quad d=-30576\\ b=43120\quad d=105105\\ b=50421\quad d=-33614\\ b=50421\quad d=100842\\ b=57624\quad d=-36015\\ b=57624\quad d=96040\\ b=64666\quad d=-37758\\ b=64666\quad d=90741\\ b=90741\quad d=-37758\\ b=90741\quad d=64666\\ b=96040\quad d=-36015\\ b=96040\quad d=57624\\ b=100842\quad d=-33614\\ b=100842\quad d=50421\\ b=105105\quad d=-30576\\ b=105105\quad d=43120\\ b=116280\quad d=-11951\\ b=116280\quad d=13320\\ b=117306\quad d=-6174\\ b=117306\quad d=6517\\ b=117649\quad d=0$$ Now you have to see if you can find $a$ and $c$ that will work from the geometry of isosceles triangles.