Perpendicular bisector calculator
For the segment from (0, 0) to (4, 6), the perpendicular bisector passes through the midpoint (2, 3) and is the line 2x + 3y − 13 = 0. Enter two end points to get the line of points equally far from both. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- X1
- 0
- Y1
- 0
- X2
- 4
- Y2
- 6
- Mx
- 2
- My
- 3
- Slope
- -0.6666666667
- Intercept
- 4.3333333333
The perpendicular bisector of a plane segment is the unique line that passes through the midpoint of the segment and meets the segment at a right angle. No network calls, no random sampling, and no language-model involvement—only deterministic arithmetic on the four numbers you provide.
What a perpendicular bisector is and why people search for it
In Euclidean plane geometry the perpendicular bisector of segment AB is the set of all points equidistant from A and B. Algebraically that set is a straight line: it passes through the midpoint M equals ((x1 plus x2) over two, (y1 plus y2) over two) and is orthogonal to the direction vector of AB.
If AB has slope m, the bisector has slope negative one over m when both slopes are defined; when AB is horizontal the bisector is the vertical line through M, and when AB is vertical the bisector is the horizontal line through M. Search intent clusters around phrases such as perpendicular bisector calculator, equation of the perpendicular bisector, and find the line equidistant from two points.
Unlike a pure midpoint landing, this page focuses on the full line equation; unlike a two-line parallel-or-perpendicular classifier, it builds one line from a single segment.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate perpendicular-bisector equation while checking homework, placing a symmetry axis on a sketch, or verifying that a triangle circumcenter construction is on track without installing a computer-algebra system. Coordinates may be any finite numbers suitable for ordinary CAD and classroom figures; the response coordinates and coefficients share the same unit system you already use. Results are fully deterministic: the same four numbers always yield the same midpoint, equation, and standard-form fields.
There is no language model, no external geometry service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials.
FAQ
What formula does the perpendicular bisector calculator use?
Midpoint M = ((x1+x2)/2, (y1+y2)/2). The bisector is the line through M orthogonal to AB, written as dx(x−mx)+dy(y−my)=0 and exported in slope-intercept (when defined) and standard form.
How are horizontal and vertical segments handled?
A horizontal segment yields a vertical bisector (x equals the midpoint x, null slope). A vertical segment yields a horizontal bisector (y equals the midpoint y, slope zero).
Is the perpendicular bisector calculator free?
The browser tool runs free on your device.
What is a perpendicular bisector used for?
Every point on it is equally far from the two end points, so it locates triangle circumcentres, splits a plane between two sites and places symmetry axes on a sketch. It is also the classic compass-and-straightedge construction.
What happens if both end points are the same point?
The calculator rejects it, because a single point has no direction and therefore no unique bisector. Enter two distinct end points to get a line.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/geom/perpendicular-bisector \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"x1":0,"y1":0,"x2":4,"y2":6}'const res = await fetch("https://api.kit.forhosting.com/geom/perpendicular-bisector", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"x1": 0,
"y1": 0,
"x2": 4,
"y2": 6
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/perpendicular-bisector",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"x1": 0,
"y1": 0,
"x2": 4,
"y2": 6
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/perpendicular-bisector", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"x1":0,"y1":0,"x2":4,"y2":6}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"x1":0,"y1":0,"x2":4,"y2":6}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/perpendicular-bisector", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"x1": 0,
"y1": 0,
"x2": 4,
"y2": 6
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.perpendicular_bisector",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Every point on that line is equidistant from the two endpoints, which is why the same construction appears in triangle circumcenters, Voronoi edges, and compass-and-straightedge proofs. This capacity accepts four finite numbers—x1, y1, x2, y2—rejects coincident endpoints and incomplete coordinates with invalid_input, and returns the midpoint, a primary equation string (slope-intercept when the bisector is non-vertical, otherwise x equals a constant), standard form coefficients Ax + By + C = 0, and orientation flags.
What a perpendicular bisector is and why people search for it
Delivering the midpoint, equation string, coefficient triple, and orientation flags as JSON makes the result easy to diff in continuous integration, feed into a larger pipeline, or copy into notes. The construction fails only when the endpoints coincide, because then there is no unique direction and no unique bisector.
How this calculator validates input and builds the line equation
When the segment is not horizontal, the bisector has a finite slope equal to negative dx over dy and a y-intercept recovered from the midpoint; those become the slope-intercept string. When the segment is horizontal the bisector is vertical and the primary equation is x equals mx with null slope fields. Send a JSON body with the four required fields x1, y1, x2, and y2. Each field must be a finite number (JSON numbers or numeric strings are accepted). The engine rejects missing fields, empty strings, NaN, and infinities with invalid_input so those failed calls are not charged. After parsing, if both coordinates match so the points coincide, the request also fails with invalid_input naming that a perpendicular bisector requires two distinct points. On success the midpoint is computed by averaging the endpoints, the segment direction is (dx, dy) equals (x2 minus x1, y2 minus y1), and the bisector is written in point-normal form as dx times (x minus mx) plus dy times (y minus my) equals zero. That expands to standard form A x plus B y plus C equals zero with A equals dx, B equals dy, and C equals minus (dx times mx plus dy times my), then normalized so A is non-negative and integer coefficients are reduced by their greatest common divisor when all three are whole numbers. Results are rounded to a stable decimal precision so the same coordinates always yield the same JSON across JavaScript engines. There is no trigonometry beyond the reciprocal-slope rule, no matrix inversion, and no external geometry service—only addition, subtraction, multiplication, and division on a Worker isolate.
Practical use, limits, pricing, and delivery channels
Each successful request is one billable item at $0.002; failed validation does not consume balance. Wire the endpoint into circumcenter demos, contest answer checkers, Voronoi edge helpers, or preprocessing steps where you already prefer an explicit midpoint-plus-orthogonal-line construction over opaque black-box widgets that cannot explain why coincident endpoints are rejected. If you need only the midpoint of the same two points, the equation of the line through the endpoints, or a classification of two given lines as parallel or perpendicular, use those dedicated landings when they exist—this capacity focuses solely on the perpendicular bisector of one plane segment.
Integration use cases
Homework and exam perpendicular-bisector drills
Students and graders confirm the midpoint and line equation from two endpoints, including clear rejection when the endpoints coincide.
Deterministic fixtures for geometry tests
Pipelines that seed unit tests or interactive segment demos can call a pure perpendicular-bisector oracle instead of reimplementing midpoint and negative-reciprocal slope in every language.
CAD, mesh, and symmetry-axis anchors
Tools that place a symmetry line, clip plane, or debug overlay through the center of an edge can recover the full bisector equation as structured JSON for import.
Technical questions
How do I compute a perpendicular bisector with this API?
Send a JSON body with x1, y1, x2, and y2 to geom.perpendicular_bisector. The response returns the midpoint, equation string, standard form, and orientation flags.
What happens if the two endpoints are the same?
The request fails with invalid_input. A unique perpendicular bisector requires two distinct points; coincident endpoints are not accepted.