Would my answer be correct?

Question

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?

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

There is an error in line 1; segments AB and BD are given to be congruent. 

There is an error in line 2; segment BE should be a perpendicular bisector. 

There is an error in line 4; segment BE is not a shared side. 

The proof is correct. I say option 2 but I am not sure please correct me!

Answer 1

To find the circumcenter of a triangle, it is necessary to find the point of intersection of the perpendicular bisectors of at least two of the sides of the triangle. That does mean that step 2 should mention CE as a perpendicular bisector of side AD. Likely there should be an additional step in the proof. How do you know that the perpendicular bisector of side AD passes through B? The reason is that a point lies on the perpendicular bisector of a segment if and only if that point is equidistant from the endpoints of the segment. That is the case, since AB=BD.

Here is a Desmos graph you might want to visit to see circumcircle of a triangle in action. Click the toggle perpendicular bisector links to see the perpendicular bisectors (and if you show BC perpendicular bisector you will also see the circumcircle)

desmos.com/calculator/pov5zr1c7g