# Degenerate cases and necessary not-sufficient conditions

- Author:
- Zoltán Kovács

For a circle with center

*A*through*B*and a free point*P*, we consider the point*D*intersection of the circle and the line*PA*. When are the lines*AD*and*BD*perpendicular?We obtain the answer by typing

`LocusEquation[ArePerpendicular[a,d],P]`

(here the result can be displayed by clicking on the right check box). We get the line *AB*passing through*A*and*B*includes three interesting types of different situations:- For
*P*in*AB*different from*A*and*B*we obtain the degenerate case in which the intersection point*D*coincides with*B*and hence the property is trivially true. - Something similar happens if
*P*is equal to*A*. This is again a degenerate case since the line*PA*does not really exist, and hence the point*D*is undefined. - Something very different, however, happens when
*P*is equal to*B*. In this case*D*is the "opposite" point to*B*along the diameter*AB*, and hence line*AD*is not perpendicular to line*BD*but equal and hence the property is false. It is important to emphasize that this condition*P*=*B*is a particular case of the locus set of necessary conditions that is not sufficient.