Point D lies inside an isosceles triangle ABC with base AC, and AD = CD. Prove that angle DBA = angle DBC

The condition says that we have an isosceles triangle in front of us.
Construct line BD by connecting points B and D.
Since AD = DB, we can conclude that AC is the median lowered to the base of an isosceles triangle, since it divides the base in half.
Let’s recall the main property of an isosceles triangle: the median dropped to the base is also the bisector and height.
It follows from this property that AC is a bisector that divides angle B into two equal angles.
Thus, the angles DBA and DBC are equal;