Certainty Model

Every point, line and circle in a figure is drawn according to how well what the program says pins it down. There are three cases.

The three states

Fixed — exactly one place

The program says enough that there is only one place it can be.

Drawn as: a crisp dot with a clean circle.

click to pan & zoom

point a at 0 0 point b at 4 0 point c at 0 3 segment a b segment b c segment c a

Free — anywhere along a line, a circle, or the plane

The program leaves it room to move: it could be anywhere along a line or a circle, or anywhere at all.

Drawn as: a dot with a wavy circle; segments as squiggly lines.

click to pan & zoom

triangle t with a b c distance between a and b is 3 distance between b and c is 4

Here c is 4 from b, but nothing says in which direction — it could be anywhere on a circle of radius 4 round b.

Either — a few separate places

The program says enough to rule out sliding, but more than one separate place still fits.

Drawn as: a dot with a jagged circle in amber, once for each place; segments as jagged lines.

click to pan & zoom

line l through (3, 0) (3, 1) point a at 0 0 point b on l distance between a and b is 5

b is on l and 5 from a, and that is true in two places: above the x-axis and below it.

Why this matters

A fixed figure means you can trust every coordinate. A free one means the program does not say enough yet — add something to pin it down. A figure with several places means it says enough, but the geometry allows more than one answer; say something that rules the others out (for instance, which side of a line a point is on, by placing a point nearby).

How it carries

• A point worked out from two others is only as fixed as they are: if either is free, so is it.
• A point with only one distance to something is always free — it can swing round at that distance.