HomeCalculatorsMathGeometryLine Intersection Calculator
Math calculator

Line Intersection Calculator

Intersection of two plane lines, each given by two points. Parallel distinct lines are empty. Coincident lines are infinite. Degenerate points are invalid. Same engine as Coordinate Geometry. Not a second engine. Not CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 11/11 tests · Production surface contract 3/3 · v1.4.0
Input interpretation
Enter values to calculate.
Result
—
Model
The intersection of two plane lines, each given by two distinct points.
Scope
Euclidean plane
Verification
Engine tested · 11/11 tests · Production surface contract 3/3 · Specification checked · v1.4.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
5 golden · 1 boundary · 5 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: STALE · Last attested schema matched 1.4.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

DeterminantΔ = A₁B₂ − A₂B₁
Pointx = (B₁C₂ − B₂C₁)/Δ, y = (C₁A₂ − A₁C₂)/Δ
Line from two pointsA = y₂ − y₁, B = x₁ − x₂, C = x₂ y₁ − y₂ x₁
iSame engine as /coordinate-geometry. Two-point line stays on /line-equation. Point-to-line distance stays on /point-line-distance. Not CAS.

How to use

1

Enter the first line as two distinct points

(0, 0) and (2, 2) is y = x.

2

Enter the second line as two distinct points

(0, 2) and (2, 0) is y = 2 − x. They meet at (1, 1).

Example calculations

Common configurations with formula and result.

ϟ

Crossing

(0, 0)–(2, 2) and (0, 2)–(2, 0)

Δ ≠ 0
(1, 1)
ϟ

Parallel

(0, 0)–(2, 0) and (0, 1)–(2, 1)

Δ = 0
∅
ϟ

Vertical ∩ horizontal

(2, 0)–(2, 5) and (0, 3)–(4, 3)

x = 2, y = 3
(2, 3)

Line Intersection calculator specification

Version 1.4.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Two lines A₁x + B₁y + C₁ = 0 and A₂x + B₂y + C₂ = 0 meet at one point when A₁B₂ − A₂B₁ ≠ 0, are parallel and empty when that determinant is 0 and the lines differ, and coincide when they are the same line. This page is mode=intersect on math.geometry.coordinate. Each line is two distinct points. Not a second engine. Not Pythagorean. Not CAS.
What it calculates
The intersection of two plane lines, each given by two distinct points.
Inputs
  • x1
  • y1
  • x2
  • y2
  • x3
  • y3
  • x4
  • y4
Outputs
  • status
  • x
  • y
  • standard1
  • standard2
Formula
A₁x + B₁y + C₁ = 0 ∩ A₂x + B₂y + C₂ = 0
Assumptions
  • Euclidean plane
  • Two distinct points per line
  • Not 3-D / spherical
  • Not Pythagorean
  • Not CAS
Units
  • length (consistent units)
Boundary conditions
  • missing coordinates → MISSING_REQUIRED_INPUT
  • coincident points on a line → INVALID_INPUT
  • parallel distinct → status=empty
  • same line → status=coincident
  • unknown mode (including cas, pythagorean) → INVALID_MODE
Example
x1=0 y1=0 x2=2 y2=2 x3=0 y3=2 x4=2 y4=0 → (1, 1)
Validation cases

2 published on this page · 11/11 tests · Production surface contract 3/3 · View evidence

  • x1=0 y1=0 x2=2 y2=2 x3=0 y3=2 x4=2 y4=0 → (1, 1)
  • mode=cas x1=0 y1=0 x2=2 y2=2 x3=0 y3=2 x4=2 y4=0 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.4.0

Other calculators in this family: Angle Between Lines Calculator, Circle Calculator, Cone Calculator, Conic Section Calculator, Coordinate Geometry Calculator, Cross Product Calculator, Cylinder Calculator, Dot Product Calculator . Explore all Geometry.

→ Angle Between Lines Calculator Acute angle between two plane lines, each given by two points. Parallel is 0°. Perpendicular is 90°. Degenerate points are invalid. Same engine as Coordinate Geometry. Not a second engine. Not CAS. Runs locally. › → Circle Calculator Compute diameter, circumference, and area from r, or arc and sector from a central angle. Not the Pythagorean page. Runs locally. › → Cone Calculator Right circular cone V = (1/3)πr²h and slant l = √(r²+h²). Same engine as Solid Geometry. Not ISO 5167-5. Not CAS. Runs locally. › → Conic Section Calculator Axis-aligned ellipse, parabola, and hyperbola in standard form. Discovery /ellipse, /parabola, /hyperbola. Not the circle-only page. Not coordinate geometry. Not CAS. Runs locally. › → Coordinate Geometry Calculator Distance, midpoint, and slope of two points in the plane. Line through two points is /line-equation. Point-to-line distance is /point-line-distance. Intersection is /line-intersection. Angle is /angle-between-lines. Not the Pythagorean-only page. Runs locally. › → Cross Product Calculator Vector product u × v in R³. 2-D uses uz = vz = 0. Same engine as Vector. Not a matrix. Not CAS. Runs locally. › → Cylinder Calculator Right circular cylinder V = πr²h and A = 2πr(r+h). Same engine as Solid Geometry. Not CAS. Runs locally. › → Dot Product Calculator Scalar product u · v = ux vx + uy vy + uz vz. Same engine as Vector. Not a matrix. Not CAS. Runs locally. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this a second coordinate engine?

No. It is mode=intersect on math.geometry.coordinate, the same engine as /calc/math/coordinate-geometry.

What if the lines are parallel?

Distinct parallel lines return ∅ (status=empty). The same line twice returns ∞ (status=coincident).

What if a pair of points coincides?

That pair does not determine a unique line. The engine returns INVALID_INPUT.