Nine-point circle calculator
For the right triangle (0, 0), (4, 0), (0, 3), the nine-point circle is centered at (1, 0.75) with a radius of 1.25, half the circumradius. Enter three vertices and get the circle through the side midpoints and the feet of the altitudes. 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
- 0
- X3
- 0
- Y3
- 3
- Center x
- 1
- Center y
- 0.75
- Radius
- 1.25
The nine-point circle is the unique circle that passes through the midpoints of a triangle’s three sides, the feet of its three altitudes, and the midpoints of the segments from the orthocenter to each vertex. Given three Cartesian vertices, this calculator builds the medial triangle from those side midpoints, finds its circumcenter (the nine-point center), and reports the radius as the distance from that center to any midpoint—exactly half the circumradius of the original triangle.
How the medial triangle yields the nine-point circle
Label the vertices A(x1, y1), B(x2, y2), and C(x3, y3). The midpoints of the sides are Ma = ((x2+x3)/2, (y2+y3)/2), Mb = ((x1+x3)/2, (y1+y3)/2), and Mc = ((x1+x2)/2, (y1+y2)/2). These three midpoints form the medial triangle, which is similar to the original triangle and half its linear scale.
The nine-point center N is defined as the circumcenter of Ma, Mb, and Mc: the unique point equidistant from the three midpoints. Solving the perpendicular-bisector intersection of MaMb and MaMc produces a closed-form 2×2 linear system whose determinant is proportional to the area of the medial triangle—and therefore nonzero precisely when ABC is non-degenerate. The nine-point radius n is the common distance from N to any midpoint; classically n = R/2 where R is the circumradius of ABC.
Equivalently, N is the midpoint of the segment joining the orthocenter H and the circumcenter O of ABC, so the Euler line always contains H, N, G (centroid), and O in fixed ratios.
A classic check is the right triangle with vertices at (0, 0), (4, 0), and (0, 3): the hypotenuse runs from (4, 0) to (0, 3), the circumcenter is at the midpoint of that hypotenuse (2, 1.5), the orthocenter sits at the right angle (0, 0), and the nine-point center is their midpoint (1, 0.75) with radius 1.25. Because the path uses only addition, subtraction, multiplication, division, and a square root for the radius, it is deterministic, O(1), and free of trigonometric libraries.
What you send and what you get back
Provide six required finite numbers: x1 and y1 for the first vertex, x2 and y2 for the second, and x3 and y3 for the third. After parsing, the service evaluates the shoelace twice-area sum; if it is exactly zero the three points are collinear and the error message states that the area is zero and asks for non-collinear vertices. Otherwise it builds the three midpoints, solves for the circumcenter of the medial triangle, measures the radius, and checks that every result is finite.
Example checks: zero, zero, four, zero, zero, three yields center (1, 0.75) and radius 1.25; zero, zero, one, one, two, two fails as collinear; a missing y3 fails naming that field. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts when you need bulk evaluation of coordinate-based nine-point circles without reimplementing the collinear gate in every language.
Where nine-point circles show up in real work
Secondary-school and college analytic geometry courses constantly ask students to locate the nine-point center and radius given three plotted points; a trusted calculator with a strict collinear check speeds homework grading and removes spreadsheet mistakes where someone confuses the nine-point center with the centroid, orthocenter, or circumcenter.
Drafting, surveying, and simple GIS workflows that already project points into a local Cartesian plane sometimes need classical triangle centers for construction layouts, Euler-line overlays, and teaching figures—having a shared numeric oracle keeps field notes and office scripts aligned on the same medial-triangle path. Game, graphics, and CAD tutorials often store mesh faces as triples of 2D screen or world coordinates; the nine-point answer feeds geometry diagnostics, midcircle-based rigs, and documentation figures without building a full computational-geometry stack.
If you later need the orthocenter alone, the circumcenter alone, the incenter, the centroid, or triangle area from the same vertices, treat those as separate search intents rather than overloading this nine-point landing.
What you can do with it
Homework and exam checks
Locate the nine-point center and radius from plotted vertices and reject collinear triples with a clear error instead of a silent nonsense circle.
Euler-line construction layouts
From three projected corner coordinates, recover the midcircle center for construction notes, Euler-line overlays, or teaching diagrams.
FAQ
What is the nine-point circle?
It is the unique circle that passes through the midpoints of a triangle’s sides, the feet of the altitudes, and the Euler points.
What method does the calculator use?
It finds the circumcenter of the medial triangle formed by the three side midpoints. The radius equals half the circumradius of the original triangle.
What happens if the three points are collinear?
Collinear vertices give zero area and do not define a triangle, so the nine-point circle is undefined.
How is the nine-point center related to the orthocenter and circumcenter?
It is the midpoint of the segment joining the orthocenter H and the circumcenter O. On the Euler line the order is H, N, G, O with HN = NG = GO/2 (fixed ratios).
What does it cost?
The browser tool is free and runs locally.
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/nine-point-circle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"x1":0,"y1":0,"x2":4,"y2":0,"x3":0,"y3":3}'const res = await fetch("https://api.kit.forhosting.com/geom/nine-point-circle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"x1": 0,
"y1": 0,
"x2": 4,
"y2": 0,
"x3": 0,
"y3": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/nine-point-circle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"x1": 0,
"y1": 0,
"x2": 4,
"y2": 0,
"x3": 0,
"y3": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/nine-point-circle", 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":0,"x3":0,"y3":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"x1":0,"y1":0,"x2":4,"y2":0,"x3":0,"y3":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/nine-point-circle", 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": 0,
"x3": 0,
"y3": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.nine_point_circle",
"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
You send six finite numbers x1, y1, x2, y2, x3, and y3; the service validates that the points are non-collinear, returns center_x, center_y, and radius with the normalized vertices, and tags method as medial_triangle_circumcenter. Collinear triples fail with invalid_input instead of a silent nonsense circle. No network calls, no random sampling, and no floating-point surprises beyond ordinary IEEE-754 arithmetic on the six numbers you provide.
How the medial triangle yields the nine-point circle
Large coordinates can still overflow IEEE-754 double range; when intermediates are non-finite the request fails with a clear overflow message instead of returning Infinity in JSON.
What you send and what you get back
Short aliases ax, ay, bx, by, cx, cy and x_a through y_c are accepted so form labels can stay compact without diverging handlers. Missing fields, empty strings, non-numeric text, NaN, and Infinity are rejected with invalid_input and are not charged. On success the JSON response returns center_x, center_y, and radius rounded for stable serialization, echoes the six normalized coordinates, and sets method to the string medial_triangle_circumcenter so logs and teaching UIs can document the algorithm. API pricing is the published base of $0.002 per successful item.
Where nine-point circles show up in real work
Because this capability is a pure parse engine, failed validation does not consume balance and successful runs stay at $0.002 per item.
Integration use cases
Game and CAD triangle helpers
Turn three 2D vertex pairs into the classical nine-point circle for mesh diagnostics, using the same path in browser and API.