I have a method, which I think is relatively simple.

Take any point on the X axis that is different from the focus, C connects with A, and make a circle with AC as the radius, which must pass through point B;

Then take point B as the center, make a circle with a radius equal to AC, and cross it on the X axis, which is point D,

It should have two points, which you need to judge. The point on the right connects a, and ABCD is a diamond. It is not difficult to prove that it is all radius.