Ic = ib, ∴ According to the combination of three lines, ∠EBC=∠ECB, ∠IBC=∠ICB,
Let ∠ ABI = X, ∠ IBC = M. ∴∠ DIC = (90-m).
∵∠acb=2∠abc,∴∠ace=∠ecb=∠ebc=(x+m)
∵AC=AI,∴∠AIC=∠ACI=(2x+m)
First prove △ACE∽△ABC, and get the square of AC =AE×AB.
∴ the square of AI =AE×AB, and it is proved that △ AIE ∴△ ABI.
∴∠AIE=∠ABI=x，
∵∠DIC+∠AIC+∠AIE= 180，
∴ (90m) +(2x+m)+x = 180, x=30,
That is ∠ ABI = 30.