# The Straight Line and the circle

## (Note: All the images are interactive, move the points around to see. Dynamic mathematics interactive web page with Cinderella )

A moving point describes a straight line when it passes from one

position to another along the shortest possible path. A straight line

can be drawn with the help of a ruler; when a pencil runs along the

edge of a ruler it leaves a trace on the paper in the form of a

straight line.

When a point moves on a surface at a constant distance from another

fixed point on the same surface it describes a circle. Because of this

property of the circle we are able to draw a circle with the help of

compasses.

The straight line and the circle are the simplest and at the same time

the most remarkable curves as far as their properties are concerned.

You are no doubt more familiar with these two curves than with

others. But you should not imagine that you know all of the most

important properties of straight lines and curves. For example, you

may not know that if the vertices of the triangles $ABC$ and $AB’C’$

lie on three straight lines intersecting at the point $S$ (Fig. 1),

the three points of intersection $M$, $K$, $L$ of the corresponding

sides of the triangles, the sides $AB$ and $A’B’$, $BC$ and $B’C’$,

and $AC$ and $A’C’$, must be collinear, that is, they lie on a single

straight line.

(Note: this image below is interactive, move the points to see the dynamic change!)

You are sure to know that a point $M$ moving in a plane equidistantly

from two fixed points, say $F_1$, and $F_2$, of the same plane, that

is, so that $MF_{1}= MF_{2}$, describes a straight line (Fig. 2).

But you might find it difficult to answer the question:

What type of curve will point $M$ describe if the distance of $M$ from

$F_1$, is a certain number of times greater than that from $F_2$ (for

instance, in Fig. 3 it is twice as great)?

The curve turns out to be a circle. Hence if the point $M$ moves in a

plane so that the distance of $M$ from one of the fixed

points. $F_{1}$ or $F_{2}$, in the same plane is always proportional

to the distance from the other fixed point, that is

$$

MF_{1} = k \times MF_{2}

$$

then $M$ describes either a straight line (when the factor of

proportionality is unity) or a circle (when the factor of

proportionality is other than unity).

This is a post to create interactive mathematics elements using Cinderella a Free Software alternative to GeoGebra which is no longer a Free Software. The files have been exported from Cinderella at html interactives)