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### Finding a Locus of Points from a Description

```Date: 03/30/2004 at 12:35:54
From: Rene
Subject: locus in geometry

I am trying to help my son figure out these three questions.  We have
asked several people and no one can help us.  Is there a formula to
use?  I know nothing when it comes to geometry, so hope you can help.

They all have to do with the locus in geometry.

1. What is the locus of points in the plane of an angle that are
equidistant from the sides of angle?

2. What is the locus of points in space that are equidistant from
two parallel planes?

3. What is the locus of points in a plane that are equidistant from
points A and B in the plane?

```

```
Date: 03/30/2004 at 13:03:47
From: Doctor Peterson
Subject: Re: locus in geometry

Hi, Rene.

This isn't a matter of formulas; I'm not sure exactly what is the key
to recognizing a locus.  Each problem is a little different.  In the
hardest cases, you have to use analytic geometry, and only when you
get an equation will you see what the locus actually is.  In many
cases, you just "see" it; the answer is obvious when you look at it
the right way.  These problems are in between; they're obvious to me
because I know enough geometry, but you can figure them out visually
or analytically if you don't immediately see it.

First I'd like to point you to some discussions of what locus is, and
some examples:

Locus
http://mathforum.org/library/drmath/view/55121.html

The Meaning of Locus
http://mathforum.org/library/drmath/view/62350.html

Locus and Equations of Lines
http://mathforum.org/library/drmath/view/52853.html

1. What is the locus of points in the plane of an angle that are
equidistant from the sides of angle?

I would start by drawing a picture:

/
+   P
/   o
/    |
o-----+--
O

I've drawn an angle, and a point P that is equidistant from the sides
of the angle (that is, the same distance from each ray). I drew in (or
would if I could draw!) perpendicular segments from P to each ray, to
remind myself how to find the distance from a point to a line (or
ray).  The locus is not yet visible; it will be the set of all places
where P might be ("locus" is just the Latin word for "place"), or
equivalently the set of all points P that fit the definition.

Now what you do next will depend on your thinking style.  If you can
visualize easily, imagine moving P around while keeping the two
distances the same; you'll find that it slides along in a particular
direction, tracing out a certain line.  If you prefer experimenting
with reality rather than picturing things in your mind, try finding
several points that fit the description, and look for what they have
in common.  Then you can fill in between, to locate all possible
points that work.  On the other hand, if you like theory and have an
analytical mind, you can just look for a theorem or two that will
determine where P can be; in this case, try drawing in segment OP and
you'll notice a pair of congruent triangles.  What do they tell you

Whatever method you use to discover the locus, or just guess at it,
you need to convince yourself that your answer is correct, either with
a real proof, or just enough thinking (along the lines of the
analytical approach I listed last) to feel confident that you've got
it right.

So it takes a certain amount of "playing" and imagining to find the
answer, and a certain amount of familiarity with theorems or facts of
geometry to see that you have it right; locus problems test your
knowledge and a bit more.  But usually if you just think about the
meaning of the terms in the problem (what is the distance from a
point in space to a plane, for example?) and think concretely, you
should be able to get it.

If you need more help, please write back and show me how far you got

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/

```

```
Date: 04/05/2004 at 11:33:22
From: Rene
Subject: Thank you (locus in geometry)

Thank you for your prompt and excellent help!  We thought about the
questions and hopefully got them right.
```
Associated Topics:
High School Definitions
High School Euclidean/Plane Geometry

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