Bisectors the easy way

Using the normal form of a straight line equation, it is easy to write down the equation of the two bisectors to two lines.

Here are the three intermal bisectors of three lines giving the incenter of this triangle. The bisectors are in red and the lines of the triangle in black.

Proof that the bisectors concur at the incenter,

Hence the bisector on the left is zero when the other two are zero, so the third bisector goes through the intersection of the first two.

This fundamental property of geometry became a simple identity in algebra.