Proof: because be and ce are divided into ABC and ACB respectively.

So angle EBC= angle ABE= 1/2 angle ABC.

Angle BCE= 1/2 Angle ACB

Because angle ABC+ angle ACB+ angle BAC= 180 degrees.

Because the angle BAC=60 degrees

So EBC angle BCE angle =60 degrees.

I think your angle EBC+ BCE+ BEC= 180 degrees.

(1) So BEC angle = 120 degrees.

Because AD bisects the angle BAC

So angle BAE= angle DAC

Because the angle BAC=60 degrees

Angle BDC= 120 degrees

So angle BAC+ angle BDC= 180 degrees.

So points A, B, D and C are four * * * cycles.

So angle ADB= angle ACB=2 angle CBE.

Angle DBC= angle DAC

Because angle bed = angle BAE+ angle ABE

Angle DBE= Angle EBC+ Angle DBC

Because Angel Abbe = Angel ·EBC (confirmed)

So angle DBE= angle DEB

(2) So DB=DE

(3) So the angle ADB=2 Angle BCE