Let's begin with triangle ABC, with the line through B and C
being a, the line through A and C being b, and the line
through A and B being c, as shown at left.
The equation of each side can be written as a dot product equation.
For example, let u be a unit vector perpendicular to a. Using GeoGebra
notation, the equation of this line is

x(u) x + y(u) y = x(u) x(B) + y(u) y(B)

where x(u) is the x-coordinate of vector u, and the others are defined

We want to have the side line equations written as

x(u) x + y(u) y - x(u) x(B) - y(u) y(B) = 0

Select any two of the side line equations, say, a and b. Add these two equations
in the form above term by term. This results in another linear equation which
must go through point C, since C is on side a and b.

It just so happens that this new line with the form a - b = 0 is the angle bisector of C.