Suppose we take a look at the triangle IaIbA and the lines AB and AC.
The line AIa divides the angle between AB and AC in two equal angles.
The line AIb divides the angle between AC and BA in two equal angles.
This means that the 2 dividing lines AIa and AIb are perpendicular.
In other words, A is the foot of an altitude of triangle IaIbIc.