Before we can even ask whether three points lie on a line, we must be sure of every mark on the page: what a coordinate is, what an arrow between points means, what "steepness" measures, and why an area can shrink to nothing. Let's build each from zero, in the order they depend on each other.
The picture. Imagine a sheet of graph paper. Draw one horizontal line (the x-axis) and one vertical line (the y-axis). Where they cross is the origin (0,0). To find (3,2) you walk 3 steps right, then 2 steps up. That final spot is the point.
Why the topic needs it. Collinearity is about where points are. Without an address for each point, "on the same line" is just a feeling. Coordinates turn that feeling into numbers we can compute with.
The picture. Three points A, B, C each need their own two numbers. Rather than invent six unrelated letters, we reuse x and y and tag them: A=(x1,y1), B=(x2,y2), C=(x3,y3). The tag tells you which point the coordinate belongs to.
Why the topic needs it. The collinearity formula x1(y2−y3)+x2(y3−y1)+x3(y1−y2) juggles three points at once. Subscripts keep their coordinates from getting mixed up.
Why subtract, and why that order? Subtraction answers the question "how much do I change?". To go from A to B your x changes by "B's x minus A's x", i.e. x2−x1. End-point minus start-point — always. If the answer is positive you moved right; negative means you moved left; zero means you didn't move sideways at all.
The picture. From A(1,1) to B(4,3): the horizontal change is 4−1=3 (3 right), the vertical change is 3−1=2 (2 up). The arrow leans up-and-to-the-right; its coordinate label is (3,2).
Why the topic needs it. The parent note builds the area from two arrows leaving the same point: AB and AC. Those arrows are how we compare directions — the heart of "same line or not".
Why a ratio (division)? We want a single number that captures "steepness" no matter how big the trip is. Going 1 up per 1 right is the same steepness as 3 up per 3 right — both give 1. Dividing rise by run cancels the size of the trip and leaves only the tilt. That is exactly why division, not subtraction, is the right tool here.
The picture. A line climbing gently (slope near 0) is almost flat; slope 1 is a 45∘ climb; a steep line has a large slope. A line going down as you move right has a negative slope.
Why we cross-multiply the slope equation. Writing x2−x1y2−y1=x3−x2y3−y2 and multiplying both sides by both denominators gives
(y2−y1)(x3−x2)=(y3−y2)(x2−x1),
which contains no division — so even a zero run causes no crash. See Slope of a Line for the full treatment.
The picture. Start with a fat triangle and slowly slide its top vertex down toward the base line. The enclosed region gets thinner and thinner until, at the instant the three corners line up, it has zero width and zero area.
Why the topic needs it. This is the whole strategy: collinear ⟺ zero area. The parent's formula
Area=21∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣
comes from measuring the arrows AB and AC (see Area of Triangle using Coordinates and Vectors and Cross Product). We build that formula in detail in the derivation deep dive.
Why area uses it. The raw expression x1(y2−y3)+… can come out positive or negative depending on whether you named the corners clockwise or anticlockwise. Area is a size, so we strip the sign with ∣⋅∣.
Why the topic needs it. It is what lets us test. Because the link runs both directions, computing the formula and getting 0 is not just a symptom of collinearity — it proves it. A one-way arrow would not be enough.
