The Point — Where Everything Begins

Every map, every GPS, every screen you have ever looked at is built from points. A point has no size, no width, no length — just location. Just here. Everything in geometry grows from this one idea.

---

Act 1 — A Point

Here is a point. It has no size. It is not a dot — a dot has size. A point is pure location.

Drag it anywhere.

Canvas 400 480
Grid 0
Scale 60
Point A at 0, 0
Drag A
Text at -2.8, -3.5 gray "👆 Drag the point anywhere"

Notice: it goes wherever you put it. It has no shape. No dimension. Just here.

---

Act 2 — Where Exactly?

Now ask yourself: where is the point?

You moved it somewhere — but how would you describe that location to someone else? You need a reference. A fixed "from here."

We draw two lines crossing at zero. One going left-right, one going up-down. Now every point in the plane has an address — two numbers, written as (x, y).

• x tells you how far left or right from the centre.
• y tells you how far up or down from the centre.

Drag the point. Watch the address change.

Canvas 400 480
Grid 1
Scale 60

# Two hidden points define the axes as Lines
Point O at 0, 0
Point xR at 4, 0
Point yT at 0, 4
Line xAxis through O xR
Line yAxis through O yT
Hide xR
Hide yT
Hide xAxis
Hide yAxis

# The draggable point
Point A at 2, 2
Drag A
Show coords A inline

# Perpendicular drops onto each axis — live, right-angle auto-marked
Perpendicular Ax from A to xAxis
Perpendicular Ay from A to yAxis
Color _perp_Ax orange
Color _perp_Ay orange
Style _perp_Ax dashed
Style _perp_Ay dashed
Show length _perp_Ax inline
Show length _perp_Ay inline

Toggle "Show x and y distances" Ax Ay _perp_Ax _perp_Ay

Press "Show x and y distances" to see exactly how each number is measured — a perpendicular drop to each axis.

This is called a reference frame. Without it, location has no meaning. You can only say where something is if you have a fixed "from here." This is true in geometry. It is true in physics. It is true in life.

---

Act 3 — Two Points, and Something New

Now bring in a second point.

The moment two points exist, something appears between them — something you cannot avoid. Distance.

You did not create it. It was always there, waiting. And now that both points have addresses, we can measure it precisely.

Drag either point. The distance updates live.

Canvas 400 480
Grid 1
Scale 60

Point A at -2, 2
Point B at 2, -1
Drag A
Drag B
Show coords A inline
Show coords B inline
Segment AB from A to B
Show length AB inline
Text at -2.8, -3.5 gray "👆 Drag A or B"

Distance is the first idea that geometry is built on. Every theorem, every construction, every shape you will ever meet — is ultimately a statement about distances.

---

What Comes Next

Two points give you distance. Add a third point — not on the same line as the first two.

Now you have something new. Direction. Space. The first shape.

That is the next lesson.