Geometry · Coordinates

Analytic Geometry in the Plane Calculator

Analyze three Cartesian points using vectors, distance, midpoint, slope, a standard-form line equation, triangle area, and a collinearity test.

Formula shown Runs locally Reviewed Aug 11, 2026

Equation guide

A(x₁,y₁) and B(x₂,y₂) define segment AB and its line; C(x₃,y₃) completes triangle ABC and tests collinearity

Points and distance

A=(x₁,y₁), B=(x₂,y₂)AB=√[(x₂−x₁)²+(y₂−y₁)²]

Line through A and B

αx+βy+γ=0α=y₂−y₁, β=x₁−x₂γ=x₂y₁−y₂x₁

Third point C

C=(x₃,y₃)2K=|det(AB,AC)|K=0 ⇔ A,B,C collinear

Geometry · Cartesian plane

Connect points, lines, and triangle area

Point A

x₁ · x-coordinate of Ay₁ · y-coordinate of A

Point B

x₂ · x-coordinate of By₂ · y-coordinate of B

Point C

x₃ · x-coordinate of Cy₃ · y-coordinate of C

Inputs and displayed decimal results use at most three decimal places. Exact internal values are retained during calculation.

AB = √[(x₂−x₁)²+(y₂−y₁)²]

A(x₁,y₁) and B(x₂,y₂) define AB; C(x₃,y₃) completes triangle ABC.

Using the model

Symbols, assumptions and limitations

Interpretation notes

A=(x₁,y₁) and B=(x₂,y₂) are distinct points that define segment AB and its line. C=(x₃,y₃) is the third point used to form triangle ABC, calculate its area, and test whether all three points are collinear. The symbols α, β, and γ are coefficients in αx+βy+γ=0, not point names. Vertical lines have undefined slope but remain valid in this form.

Worked examples

See the method in practice

A 3–4 displacement

A=(0,0) and B=(3,4) give vector (3,4), distance 5, midpoint (1.5,2), and slope 4/3.

Triangle area

With C=(3,0), the determinant of AB and AC gives triangle area 6.

Questions

Frequently asked

What is point C used for?

C=(x₃,y₃) completes triangle ABC. It is used for the area and collinearity test, but it does not change the line determined by A and B.

Why is vertical slope undefined?

Its horizontal change is zero, so Δy/Δx would divide by zero.

How is collinearity detected?

Three points are collinear exactly when the determinant of vectors AB and AC is zero.

Why use standard line form?

The form αx+βy+γ=0 represents vertical and nonvertical lines uniformly; the Greek letters are coefficients, not the input points A, B, and C.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax Precalculus — The Rectangular Coordinate Systems and Graphs

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology