Enter three points to precisely determine if they are collinear
P\u2081
P\u2082
P\u2083
Result
Step-by-Step Derivation
Collinearity Formula
D = x\u2081(y\u2082\u2212y\u2083) + x\u2082(y\u2083\u2212y\u2081) + x\u2083(y\u2081\u2212y\u2082)
When |D| = 0, the three points are collinear. When |D| \u2260 0, they are not collinear. This formula comes from the triangle area formula: zero area means collinearity.
⚠Note: The three points must not be identical; otherwise, a valid line segment cannot be formed for the determination.
What Are Collinear Points?
Three points are collinear if they lie on the same straight line. In analytic geometry, there are several methods to determine collinearity, with the determinant (area) method being the most precise and reliable.
Determinant Method
When points are collinear, triangle area is 0. The determinant D = x\u2081(y\u2082-y\u2083)+x\u2082(y\u2083-y\u2081)+x\u2083(y\u2081-y\u2082) equals twice the signed area.
Slope Method
Compare slopes of two line segments. If k\u2081 = k\u2082, points are collinear. Limitation: vertical lines have undefined slopes.
Vector Collinearity
Vectors P\u2081P\u2082 and P\u2081P\u2083 are collinear if there exists \u03bb such that P\u2081P\u2083 = \u03bb\u00b7P\u2081P\u2082.
Geometric Meaning
Collinear points lie on the same line. They can be ordered in three ways (each point in the middle), but the collinear property remains unchanged.
Teaching Example: Check if A(1,2), B(3,4), C(5,6) are collinear. D = 1\u00d7(4-6) + 3\u00d7(6-2) + 5\u00d7(2-4) = -2+12-10 = 0, so they are collinear (all lie on y = x+1).
Applications
Vector CollinearityThree PointsShape VerificationAnalytic GeometryCompetition Math
Frequently Asked Questions
How to determine if three points are collinear?▼
Use the determinant method: S = |x\u2081(y\u2082-y\u2083) + x\u2082(y\u2083-y\u2081) + x\u2083(y\u2081-y\u2082)|/2. When S=0, the points are collinear. This is based on the cross product.
What is the principle behind the determinant method?▼
The triangle area = |x\u2081(y\u2082-y\u2083) + x\u2082(y\u2083-y\u2081) + x\u2083(y\u2081-y\u2082)|/2. Zero area = collinear points. Pure integer arithmetic, no slope needed.
What is the difference between slope and determinant methods?▼
Slope needs special cases for vertical lines and division may cause errors. Determinant uses pure arithmetic, making it more general and precise.
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