In the above image, it says $$AE = \frac{bc}{c+a}$$ and $$AF = \frac{bc}{a+b}$$ But $AE$ and $AF$ are tangents from $A$ to the incircle. As tangents on a circle from a given point are equal, $AE=AF$ which implies $b=c$ which is absurd. What is wrong here?
$AE$ and $AF$ are not tangents. The angle bisector is not usually perpendicular to the opposite side. If that would be the case for all sides, then you have an equilateral triangle. It's easy to see in the figure that $AD$ is not perpendicular to $BC$