ForHosting KIT · Math

Basic proportionality (Thales)

With AD of 3, DB of 6, AE of 2 and EC of 4, both ratios are 0.5, so Thales' proportion holds. Enter the four segments cut by a parallel line and check the proportion or find a missing one. Free in your browser.

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Worked example

If you enter

Ad: 3
Db: 6
Ae: 2
Ec: 4
Decimal places: 6

You get

Ratio ad db: 0.5
Ratio ae ec: 0.5

Thales BPT: AD/DB = AE/EC when DE∥BC; solve missing segment. Search intent: basic proportionality theorem. No network calls or random seeds are required. Failed jobs are not billed.

What the basic proportionality theorem says

Take a triangle ABC and draw a line DE parallel to the side BC, with D on side AB and E on side AC. The basic proportionality theorem, credited to Thales, says that this line cuts the two sides in the same ratio: AD divided by DB equals AE divided by EC. The small triangle ADE is a scaled copy of the large triangle ABC, which is why the pieces keep the same proportion. With AD of 3, DB of 6, AE of 2 and EC of 4, both ratios are 0.5, so the proportion holds.

How to use it, and the mistakes to avoid

The theorem is most often used to find one missing segment. Write the proportion, cross-multiply and divide: with AD of 3, DB of 6 and AE of 2, the missing EC is 2 times 6 divided by 3, which is 4. The most common slip is mixing two forms of the ratio, for example comparing AD with the whole side AB on one side but AE with the piece EC on the other. Another is forgetting that the theorem only applies when DE is parallel to BC. If you ask for a fixed number of decimals, each ratio is rounded once, with halves rounded up.

What you can do with it

Teaching

Pinned goldens.

FAQ

What does the basic proportionality theorem say?

If a line parallel to one side of a triangle crosses the other two sides, it divides them in the same ratio. In triangle ABC with DE parallel to BC, AD divided by DB equals AE divided by EC.

How do I find a missing segment?

Write the proportion, cross-multiply and divide. With AD of 3, DB of 6 and AE of 2, the missing EC is 2 times 6 divided by 3, which gives 4.

What does it mean when the two ratios match?

It means the proportion holds, which is exactly what happens when DE is parallel to BC; in the example both ratios are 0.5. If the ratios differ, DE cannot be parallel to BC.

Can I choose how many decimals the ratios have?

Yes. If you ask for a number of decimals, each ratio is rounded once, at the end, with halves rounded up. That keeps a ratio such as 0.5 exact and long ratios readable.

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

POSThttps://api.kit.forhosting.com/math/basic-proportionality

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/math/basic-proportionality \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"ad":3}'

const res = await fetch("https://api.kit.forhosting.com/math/basic-proportionality", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "ad": 3
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/math/basic-proportionality",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "ad": 3
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/basic-proportionality", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"ad":3}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"ad":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/basic-proportionality", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "ad": 3
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "math.basic_proportionality",
  "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

Per request$0.002

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

The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks, automation, or teaching demos.

How it works

Thales BPT: AD/DB = AE/EC when DE∥BC; solve missing segment. AD/DB = AE/EC. No network calls or random seeds are required. Failed jobs are not billed. No network calls or random seeds are required. Failed jobs are not billed. The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks, automation, or teaching demos. The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks, automation, or teaching demos.

Inputs, limits and rounding

Optional decimals use half-up once. Optional output fields are omitted, never null. No network calls or random seeds are required. Failed jobs are not billed. No network calls or random seeds are required. Provide required JSON fields with DOM-safe names. Invalid types raise invalid_input. The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks, automation, or teaching demos. The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks,

When to use it

Use when search intent matches. Thales BPT: AD/DB = AE/EC when DE∥BC; solve missing segment. EN-only delivery; Nitro hydrates locales. No network calls or random seeds are required. Failed jobs are not billed. No network calls or random seeds are required. The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks, automation, or teaching demos. The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks, automation, or teaching

Integration use cases

Quick check

Free widget.

Automation

API same JSON.

Technical questions

Free?

Yes tier A.

Network?

No.

Bad input?

invalid_input.

API?

Yes.

Locales?

EN-only; Nitro hydrates.

Read the full KIT documentation →

Run — free