Straight Lines Really Are Straight

Geometry Level 4

The line \(4x + y = 1\) passes through the point \(A(2,-7)\) and intersects point \(B\) on the line \(BC\) whose equation is \(3x - 4y + 1=0\). The equation of the line \(AC\) is \(ax + by + c = 0\) (with \(a,\) \(b,\) and \(c\) all coprime) and constructed so that the distance from \(A\) to \(B\) is the same as the distance from \(A\) to \(C\).

What is the minimum positive value of \(a+b+c\)?

Note: \(A,B,C\) are all different points in the coordinate plane.

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