Proof of Incenter divides the internal angle bisector in the ratio of b+c:a

 Incenter of triangle divides the internal angle bisector in ratio of b+c:a

For proof of BD:CD::c:b visit following links

https://maths49.blogspot.com/2022/01/angle-bisector-divides-opposite-side-in.html

Or 

https://maths49.blogspot.com/2022/01/internal-angle-bisector-divides.html