# 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.

×