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### Strange Points of Locus

```
Date: 6/6/96 at 8:24:33
From: Anonymous
Subject: Strange Points of Locus

Given two fixed points, A and B, on a plane, if P is a moving point
such that PA and PB are perpendicular and the locus of P is a circle,
should we exclude points A and B?
```

```
Date: 6/14/96 at 0:54:4
From: Doctor Brian
Subject: Re: Strange Points of Locus

Yes, we should.  P can not be the same as A.  If it were, then we'd
have the statement that AA and AB are perpendicular.

1. AA would have to be a line segment, but it's not....line segments
have positive length, not a length of zero.

Now, you might want to make the argument that this is a special
case, that AA is a limiting case of the length of all perpendicular
segments to AB through A.  The problem with that argument is that an
identical argument can be made for all segments forming an 89 degree
angle with AB.  Or any angle you want!  You can't really put a
specific degree measure on the angle.

2. Another argument for this comes from the definition of cosine in
vector spaces.  If you've ever worked with vectors, you may have run
across the definition of the cosine of the angle between two vectors:

cos theta = (the "dot product" of the vectors)/(the product of the
lengths of the vectors)

If one of the vectors here is AA, then its length is zero, and so the
denominator of the expression above would be zero.  So the formula
basically doesn't allow us to come up with an angle.

Now, you can repeat the whole thing to show why B must be eliminated
from the locus as well.

-Doctor Brian,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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