According to the definition of isosceles triangle, AB=AC is obtained.

So (4-3) 2+(2-5) 2 = (4-x) 2+(2-y) 2.

So (x-4) 2+(y-2) 2 = 10.

The trajectory equation of C is: (x-4) 2+(y-2) 2 = 10.

It is a circle with the center (4,2) as the center and the root number 10 as the radius.

I hope I can help you, please adopt it.