Straight Lines Really Are Straight

Geometry Level 4

The line 4x+y=14x + y = 1 passes through the point A(2,7)A(2,-7) and intersects point BB on the line BCBC whose equation is 3x4y+1=03x - 4y + 1=0. The equation of the line ACAC is ax+by+c=0ax + by + c = 0 (with a,a, b,b, and cc all coprime) and constructed so that the distance from AA to BB is the same as the distance from AA to CC.

What is the minimum positive value of a+b+ca+b+c?

Note: A,B,CA,B,C are all different points in the coordinate plane.

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