This topic mainly examines the comprehensive application of circle, the judgment and properties of congruent triangles, the formula of arc length and the theorem of central angle. Making auxiliary lines correctly is the key to solve the problem.

The first question uses the central angle theorem to find the angle BOD= 120 degrees, and then uses the arc length formula to find the length of the bad arc BD.

Solution: (1) connects OB and OD, because the angle BE=AB= 120 degrees, so the angle BOD= 120 degrees. The detailed answer is here/exercise/math /79946 1 as shown in the figure.

(1) If the radius of circle O is 3 and the angle DAB= 120 degrees, find the length of bad arc BD;

(2) Verification: BF =1/2bd;

(3) Let G be the midpoint of BD, and explore: Is there a point P (different from point B) on the circle O that makes PG=PF? And explain the positional relationship between PB and AE.