The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of that triangle intersect
Since point D is the circumcenter of triangle ABC, then
DE is the perpendicular bisector of AC
DF is the perpendicular bisector of BC
DH is the perpendicular bisector of AB
In triangle ADB
Since DH is perpendicular to AB ----- Proved
Since D is the midpoint of AB ------ Proved
Then triangle ADB is an isosceles triangle
Then AD = DB
The correct answer is BD
In triangle ADC
Since DE is perpendicular to AC
Since E is the midpoint of AC
Then triangle ADC is an isosceles triangle
Then AD = DC
Then the answer should be
AD = DB and DC