Angle+Bisectors+and+Trilinear+Lines

being //a//, the line through A and C being //b//, and the line through A and B being //c//, as shown at left. || For example, let //u// be a unit vector perpendicular to //a//. Using GeoGebra notation, the equation of this line is
 * [[image:trilinear_angle_bisectors1.png width="560" height="427"]] || Let's begin with triangle ABC, with the line through B and C
 * [[image:trilinear_angle_bisectors2.png width="560" height="300"]] || The equation of each side can be written as a dot product equation.

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 accordingly.

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 || 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//.
 * media type="custom" key="11468050" || Select any two of the side line equations, say, //a// and //b//. Add these two equations

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