Connect the center of the circle I with three tangent points to make a high line AH, connecting IA, IB and IC.

Let the radius of circle I be r, then ID=IE=IF=r? Let the height ah = h.

∫S△ABC = S△IBC+S△IAC+S△IAB

∫S△ABC = 1/2？ ahS△IBC= 1/2？ ar？ S△IAC= 1/2？ brS△IAB= 1/2？ chrome

∴ah=ar+br+cr=(a+b+c)r

∴h/r=(a+b+c)/a

∵IE⊥BC？ AH⊥BC

∴IE∥AH

∴AD/ID=AH/HE

∴AD/ID=(a+b+c)/a

∴AI/ID=(b+c)/a