What term is being described as the set of all points that are at a fixed distance from a fixed point in the plane?

In Mathematics we often say "the set of all points that ... ".
What does it mean?

	A set is just a collection of things with some common property.
	When we collect ALL points that share a property we can end up with a line, a surface or other interesting thing.

Points can make a line

"the set of all points on a plane that are a fixed distance from a central point".

So, just a few points start to look like a circle, but when we collect ALL the points we will actually have a circle.

Try drawing one yourself (move any point):

images/geom-locus.js

(Note: the points are drawn as dots so you can see them,
but they really should have no size at all)

Surface

Imagine this happening in 3D space: all the points that are a fixed distance from a center make a sphere!

The idea of "the set of all points that ..." is used so much it has a name: Locus.

A Locus is a set of points that share a property.

So, a circle is "the locus of points on a plane that are a fixed distance from the center".

Note: "Locus" usually means that the points make a continuous curve or surface.

Example: An ellipse is the locus of points whose distance from two fixed points add up to a constant.

So, no matter where we are on the ellipse, we can add up the distance to point "F" and to point "G" and it will always be the same result.

(The points "F" and "G" are called the foci of the ellipse)

The idea of "Locus" can be used to create some weird and wonderful shapes!

7922, 7923, 7924, 7925, 7926, 7927, 7928, 7929, 7930, 7931

Start with a point C, and collect all of the points a fixed distance r units away from it. Give this collection of points a name: circle. I have drawn a circle in Figure 17.1. The starting point is called the center of the circle. Any line segment having the center of the circle as one endpoint and any point on the circle as the other endpoint is called a radius of the circle. Because all points on the circle are a distance of r units away from the center, all radii of a circle are congruent. This will be stated as a theorem, though the proof would take no more than a line or two, with the reasons being either “given” or “definition of a circle.”

Figure 17.1A circle with center C and radius ¯AC.

Theorem 17.1: All radii of a circle are congruent.

A circle is the set of all points in a plane that are a fixed distance from a given point.

The center of the circle is the point equidistant from all points on the circle.

A radius of a circle is a line segment with one endpoint being the center of the circle, the other endpoint being a point on the circle.

Don't limit yourself to only drawing radii of circles. Circles get really interesting when you connect points on a circle. A line segment that joins two points on the circle is called a chord of the circle. A diameter of a circle is a chord that contains the center of the circle. The length of a diameter of a circle is twice the length of the radius of a circle. This can be proven by using the Segment Addition Postulate (Postulate 3.5).

If you have three points on a circle, you can connect them to form an inscribed angle. An inscribed angle of a circle is an angle whose vertex is a point on the circle and whose sides are chords of the circle. You can also construct inscribed polygons by using points on the circle as the vertices.

Two circles that coincide are congruent. In order for two circles to be congruent, the lengths of the radii must be congruent. If two circles have the same center they are called concentric circles.

There are times when you will need to measure the distance around the circle. When you did this with polygons, it was called the perimeter. With circles, it will be called the circumference of the circle.

A chord is a line segment that joins two points on a circle.

A diameter of a circle is a chord that contains the center of the circle.

An inscribed angle of a circle is an angle whose vertex is a point on the circle and whose sides are chords of the circle.

An inscribed polygon of a circle is a polygon whose vertices are points on the circle and whose sides are chords of the circle.

Congruent circles are circles that have congruent radii.

Concentric circles are circles that have the same center.

The circumference of a circle is the linear measure of the distance around the circle.

Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

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