Revisiting coordinate 3!

Geometry Level 5

A triangle \(ABC \) is given where vertex \(A\) is \((1,1)\) and its orthocenter is \((2,4)\).

Also sides \(AB\) and \(BC\) are members of the family of lines a\(x+by+c = 0\), where \(a,b\) and \(c\) form an arithmetic progression.

If the coordinates of the circumcenter of the triangle \(ABC\) can be represented as \((d,e)\), where \(d\) and \(e\) are real numbers, find \(d+e\).

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