Coordinate geometry
Plane analytic geometry: distance and midpoint, slope and the equation of a line in general, slope-intercept and two-point form, point-to-line distance, line intersection and angle, parallel/perpendicular tests, circle equations, point-in-circle position and triangle or polygon area.
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Geometry input
A line is written in general form A x + B y + C = 0; polygon vertices go in as “x,y” pairs separated by semicolons or newlines.
Result
5What this tool does
- Check analytic geometry homework: distance, midpoint, slope and all three notations of a line equation in one place.
- Decide whether two lines are parallel or perpendicular, and get their intersection and angle — with a clear message when parallel lines have no single intersection.
- Convert between the general and standard equation of a circle: feed in D, E, F to get the centre and radius, or the other way round.
- Measure the area of an irregular plot or a polygon on a drawing by listing its vertices; the shoelace formula works for concave polygons too.
Example
Input
Calculation “Polygon area (shoelace formula)”, vertices 0,0; 1,0; 1,1; 0,1
Output
Vertices 4, area 1, perimeter 4
The unit square comes out as exactly 1; listing the vertices clockwise gives the same area because the shoelace result is taken as an absolute value.
Frequently asked questions
Why does a vertical line have no slope?
Slope is defined as Δy/Δx, and a vertical line has Δx = 0, so the division is undefined — the slope mode says so explicitly. The line equation mode still works and reports the general form together with x = c.
What happens with parallel lines?
Parallel (or coincident) lines give no unique solution, so the intersection mode reports that the lines are parallel and have no single intersection. In the angle mode the angle is 0° and the parallel test returns yes.
Why would a circle radius come out negative?
The general equation x² + y² + D x + E y + F = 0 describes a real circle only when D² + E² − 4F ≥ 0, with radius √(D²+E²−4F)/2. When that discriminant is negative the equation has no real solutions and the tool reports that no such circle exists.
Does the order of polygon vertices matter?
The area is the absolute value of the shoelace sum, so clockwise and counter-clockwise both work. The vertices must still be listed in the order they appear around the boundary — a self-intersecting polygon has no meaningful area.
Are the parallel, perpendicular and collinear tests exact?
They use a relative tolerance of 1e-9: the parallel test compares |A₁B₂ − A₂B₁| against the normals’ lengths, perpendicular compares the dot product the same way, and collinear compares the coordinate area against the magnitude of the products involved. Approximate inputs such as 1/3 therefore never flip the verdict.
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