This is the specific case when angles ADC and ABC are right angles.

Again, those triangles would be right angled.Quote:

OR

AC divides BD into 2 equal parts

Those angles being equal can have the angles ADC and ABC both $\displaystyle >90^o$Quote:

OR

The angels ACB and ACD are equal ?

in which case the quadrilateral would not be cyclic,

since opposite angles would not necessarily sum to $\displaystyle 180^o$

Angles ADC and ABC either have to be right angled, or summing to $\displaystyle 180^o$

Having them sum to $\displaystyle 180^o$ is required.