Https://imgur. com/a/Z8bKMmK 10 point pls help

C is the circumcenter of isosceles triangle ABD with vertex angle ∠ABD. Does the following proof correctly justify that triangles ABE and DBE are congruent?

It is given that triangle ABD is an isosceles triangle, so segments AB and DB are congruent by the definition of isosceles triangle.
It is given that C is the circumcenter of triangle ABD, making segment BE a median.
By the definition of perpendicular, angles AEB and DEB are 90°, so triangles ABE and DEB are right triangles.
Triangles ABE and DEB share side BE making it congruent to itself by the reflexive property.
Triangles ABE and DBE are congruent by HL.