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### Locus

```
Date: 05/03/99 at 08:37:49
From: Jo
Subject: Locus

What do you have to do to get the perpendicular bisector of all the
loci of a triangle? What does it mean when it asks you about
equiddistance from certain points?

Thanks.
```

```
Date: 05/03/99 at 13:02:03
From: Doctor Peterson
Subject: Re: Locus

Hi, Jo. Thanks for writing.

Your first question doesn't make a lot of sense, because there is no
"locus of a triangle." But I think I can see what you're asking about.

A "locus" is the set of all points that satisfy some rule or
description, and a perpendicular bisector is one of the simplest
examples of a locus. If I asked you "what is the locus of all points
equidistant from two given points?", the answer would be "the
perpendicular bisector of the segment determined by the two points."
Here's what the question means:

What is the locus of all points ...

(What geometrical object consists of all points X ...)

that are equidistant from A and B?

(... for which the distances AX and BX are equal?)

That is, "equidistant from two points" means "the same distance from
both points."

The perpendicular bisector ...

(the line through the midpoint of the segment, perpendicular
to the segment)

of the segment determined by A and B.

(you make the segment by connecting A and B with a straight
line)

You can prove (and probably your text did this) that any point on the
perpendicular bisector of a segment AB is the same distance from both
endpoints; you can see this by drawing in the segments and seeing
that you have an isosceles triangle, so that AX = BX:

|
+ X
/|\
/ | \
/  |  \
/   |   \
/    |    \
/     |     \
/      |      \
/       |       \
/        |        \
/         |         \
/          |          \
/           |           \
/            |_           \
/             | |           \
+--------------+--------------+
A              |              B
|
|
|
|
|
|
|

This tells us that any point on the perpendicular bisector is
equidistant from A and B, so it is part of the locus we are looking
for.

Conversely, you can show that if any point is equidistant from A and
B, it must be on the perpendicular bisector, so there are no other
points in the locus: the perpendicular bisector IS the entire locus
we're looking for.

That's what a locus is: the set of ALL points that fit some
description (in this case, being equidistant from A and B). In our
case, the perpendicular bisector is "the locus, the whole locus, and
nothing but the locus" of equidistant points.

There are many other examples of a locus. For example, the locus of
points 1 inch from point A is the circle of radius 1 centered at A.
Every point of the circle is 1 inch from A, and every point one inch
from A is part of the circle.

I hope that helps a little to clarify the idea of a locus, and the
meaning of equidistant. If you have problems you still have trouble
with, maybe you can write back with a specific problem so I can see
the wording of it.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Definitions
High School Euclidean/Plane Geometry
High School Geometry
High School Triangles and Other Polygons

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