Degenerate cases and necessary not-sufficient conditions
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.