Collinearity of Three Points Calculator

What is Collinearity of Three Points?

Collinearity of three points is when three points lie on the same straight line. Imagine drawing a line through two points - if the third point also falls exactly on that line, then all three points are collinear!

How to Calculate Collinearity of Three Points

To check if three points are collinear, we use their coordinates and a special formula. We'll compare the slopes between pairs of points. If all the slopes are the same, the points are collinear!

Formula

We use this formula to check collinearity:

\[ (y_2 - y_1)(x_3 - x_2) = (y_3 - y_2)(x_2 - x_1) \]

Where:

• \((x_1, y_1)\) are the coordinates of the first point
• \((x_2, y_2)\) are the coordinates of the second point
• \((x_3, y_3)\) are the coordinates of the third point

Calculation Steps

1. Write down the coordinates of your three points
2. Plug these coordinates into the left side of the formula: \((y_2 - y_1)(x_3 - x_2)\)
3. Now plug the coordinates into the right side: \((y_3 - y_2)(x_2 - x_1)\)
4. Calculate both sides
5. If both sides are equal, the points are collinear!

Example

Let's check if these points are collinear: A(1, 1), B(3, 3), and C(5, 5)

1. Left side: \((3 - 1)(5 - 3) = 2 \times 2 = 4\)
2. Right side: \((5 - 3)(3 - 1) = 2 \times 2 = 4\)
3. Both sides equal 4, so these points are collinear!

Visual Representation