Medial triangle calculator
For the triangle (0, 0), (6, 0), (0, 6), the side midpoints are (3, 3), (0, 3) and (3, 0), and the medial triangle they form has area 4.5, a quarter of 18. Enter three vertices to get both triangles. 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
- 6
- Y2
- 0
- X3
- 0
- Y3
- 6
- Mx1
- 3
- My1
- 3
- Mx2
- 0
- My2
- 3
The medial triangle of a plane triangle—also called the midpoint triangle—is the triangle formed by joining the midpoints of the three sides. By the midpoint theorem each side of that inner triangle is parallel to one side of the outer triangle and exactly half as long, so the medial triangle has one fourth the area of the original figure. Students, contest solvers, CAD helpers, and geometry pipeline authors need those six midpoint coordinates and the two areas as stable numbers they can pin in tests, not only a sketch on paper.
What the medial triangle is and why people search for it
In Euclidean plane geometry the medial triangle of triangle ABC is the triangle DEF whose vertices are the midpoints of the sides: D is the midpoint of BC, E is the midpoint of AC, and F is the midpoint of AB. The Varignon-style construction for a triangle collapses to this single midpoint triangle, and the classical midpoint theorem states that each side of DEF is parallel to one side of ABC and half as long.
Consequently the three small corner triangles ADF, BDE, and CEF each share the same area as the medial triangle, and area of DEF equals one fourth of area of ABC. Search intent clusters around phrases such as medial triangle calculator, midpoint triangle, triangle midpoints of sides, and medial triangle area formula.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate medial triangle while checking homework, drawing midsegments, or verifying that a midpoint construction has area one fourth of the outer triangle without installing a computer-algebra system. Coordinates may be any finite numbers within a large absolute bound suitable for ordinary CAD and classroom figures; the response coordinates share the same unit system you already use. Results are fully deterministic: the same six numbers always yield the same medial vertices and areas.
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. If you need only the centroid, circumcenter, incenter, or side lengths, use those dedicated landings when they exist—this capacity focuses solely on the medial triangle from three distinct vertices and the quarter-area relation that follows from the midpoint theorem.
What you can do with it
Homework and midsegment triangle drills
Students and graders confirm medial vertices and the one-fourth area rule from three coordinates, including clear rejection when two vertices coincide.
FAQ
What formula does the medial triangle calculator use?
Each medial vertex is the midpoint of a side: midpoint of BC, AC, and AB.
Is the medial triangle calculator free?
The browser tool runs free on your device.
Why is the medial triangle area one fourth of the original?
By the midpoint theorem each side of the medial triangle is parallel to a side of the outer triangle and half as long, so linear scale is 1/2 and area scale is 1/4.
How long are the sides of the medial triangle?
Each side is exactly half as long as one side of the original triangle and parallel to it. That is the midpoint theorem, and it is the reason the inner triangle looks like a smaller, upside-down copy of the outer one.
What happens if two vertices are the same point?
The calculation stops straight away with a clear message. The three vertices must all be different, otherwise there is no proper triangle whose side midpoints could be joined.
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/medial-triangle \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"x1":0,"y1":0,"x2":6,"y2":0,"x3":0,"y3":6}'const res = await fetch("https://api.kit.forhosting.com/geom/medial-triangle", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"x1": 0,
"y1": 0,
"x2": 6,
"y2": 0,
"x3": 0,
"y3": 6
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/medial-triangle",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"x1": 0,
"y1": 0,
"x2": 6,
"y2": 0,
"x3": 0,
"y3": 6
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/medial-triangle", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"x1":0,"y1":0,"x2":6,"y2":0,"x3":0,"y3":6}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"x1":0,"y1":0,"x2":6,"y2":0,"x3":0,"y3":6}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/medial-triangle", 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": 6,
"y2": 0,
"x3": 0,
"y3": 6
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.medial_triangle",
"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
This capacity accepts six numbers—x1, y1, x2, y2, x3, y3—checks that the three vertices are pairwise distinct, and returns the medial vertices mx1 through my3 together with area, original_area, and a fixed area_ratio of 0.25. When any two vertices coincide the request fails immediately with invalid_input so failed validation is never charged.
What the medial triangle is and why people search for it
Delivering mx1, my1, mx2, my2, mx3, my3, area, and original_area as JSON makes the result easy to diff in continuous integration, feed into a larger pipeline, or copy into notes. Unlike circumcenter or orthocenter constructions, the medial triangle uses only averages of endpoint pairs—no perpendiculars, no angle bisectors—so the arithmetic stays elementary and fully explained for classroom and engineering use.
How this calculator validates input and builds the midpoint triangle
For accepted input the medial vertices are computed as mx1 equals (x2 plus x3) divided by two and my1 equals (y2 plus y3) divided by two for the midpoint of BC; mx2 and my2 for the midpoint of AC; mx3 and my3 for the midpoint of AB. The original area uses the absolute shoelace formula on the three input vertices, and the medial area is exactly one fourth of that value—consistent with recomputing shoelace on the three midpoints. Send a JSON body with the six required fields x1, y1, x2, y2, x3, and y3 for vertices A, B, and C. 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, it checks pairwise distinctness: if vertex one coincides with vertex two, or one with three, or two with three within a tiny floating-point epsilon, the request fails with a clear message that two vertices coincide and three distinct vertices are required for a medial triangle. That rule matches the geometric requirement for naming a triangle: you need three different corners before midpoints of sides are well-defined as a non-degenerate labeling of a figure. Results are rounded to a stable decimal precision so the same coordinates always yield the same JSON across JavaScript engines. The response echoes the six input coordinates, returns the six medial coordinates, area, original_area, area_ratio set to 0.25, and method set to medial_triangle_midpoints so callers can assert the algorithm in tests. There is no trigonometry, no matrix inversion, and no external geometry service—only averages and shoelace 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 median and midsegment demos, contest answer checkers, or preprocessing steps where you already prefer explicit midpoints of sides over opaque black-box widgets that cannot explain why two coincident corners are rejected.
Integration use cases
Deterministic fixtures for geometry tests
Pipelines that seed unit tests or interactive midpoint demos can call a pure medial-triangle oracle instead of reimplementing midpoint averages in every language.
CAD, mesh, and midsegment construction helpers
Tools that place midsegment edges, labels, or subdivision markers on a triangular face can recover the medial triangle as structured JSON for import.
Technical questions
How do I compute a medial triangle with this API?
Send a JSON body with x1, y1, x2, y2, x3, and y3 to geom.medial_triangle. The response returns mx1–my3 as the three side midpoints plus area and original_area.
What happens if two vertices are the same point?
The request fails with invalid_input because two vertices coincide and three distinct vertices are required. Those validation failures are not charged.