---
title: "Pythagorean theorem: a^2+b^2=c^2"
description: "tl;dr; If you aren't eager to learn proof of the pythagorean theorem, you can stop reading. I hope, ..."
author: "Madiyar92"
published: "2015-10-28T15:39:28+00:00"
modified: "2015-10-28T16:45:24+00:00"
locale: "ru"
canonical_url: "https://yvision.kz/post/pythagorean-theorem-a-2-b-2-c-2-621205"
markdown_url: "https://yvision.kz/post/pythagorean-theorem-a-2-b-2-c-2-621205/markdown"
site_name: "Yvision.kz"
---

# Pythagorean theorem: a^2+b^2=c^2

> tl;dr; If you aren't eager to learn proof of the pythagorean theorem, you can stop reading. I hope, ...

**tl;dr;** If you aren't eager to learn proof of the pythagorean theorem, you can stop reading.

I hope, almost everybody knows or has heard about **pythagorean theorem.** I am not aware in which year of school it is taught, but it is definitely in school syllabus. In this post, I am going to write proof of it.

![Pythagorean theorem: a^2+b^2=c^2](http://storage.yvision.kz/images/user/madiyar92/j1217GaYb2I6E81488Wv3LRP5ADI9v.png)

Let's recall pythagorean theorem. If we have right triangle(triangle with a 90-degree angle) ABC , then this equation is true:

**a^2 + b^2 = c^2**

or in other denotations:

**|BC|^2 + |AC|^2 = |AB|^2**

** **

**1. Proof by area**

**

![Pythagorean theorem: a^2+b^2=c^2](http://storage.yvision.kz/images/user/madiyar92/1UtVW3op66D1ac121WCeU80U0157MP.png)

**

We want to proof that a^2+b^2=c^2.

Area of the outside rectangle is equal to (a+b)^2. But, there is a other way to calculate outside area, we can sum up area of the inside rectangle and four right triangles, which are located on the corners.

(a+b)^2 = c^2 + (a*b)/2 * 4

if we open the bracket on the left side,

a^2+2*a*b+b^2=c^2+2*a*b

subtract 2*a*b from both sides,

a^2+b^2=c^2

**End of the proof**

** **

**2. Proof by triangle**

**

![Pythagorean theorem: a^2+b^2=c^2](http://storage.yvision.kz/images/user/madiyar92/uNuBw5HxjguZSUGM2G9CZ1TrfqI5Ee.png)

**

Try to come up, why angle at Point **C** is divided into **b** and **a** ? (btw, it is called similar triangles).

**sin(a)** = opposite/hypotenuse = |BC| / |AB| = |BD| / |BC|

**cos(a)** = adjacent/hypotenuse = |AC| / |AB| = |AD| / |AC|

in other words,

|BC|^2 = |AB| * |BD| - derived by cross multiplying the **sin(a)** proportion above.

|AC|^2= |AB| * |AD| - derived by cross multiplying the **cos(a)** proportion above.

if we sum up two equations,

|BC|^2+|AC|^2=|AB|*(|AD|+|BD|)

From the picture we can verify that:

|AD|+|BD|=|AB|

Thus,

|BC|^2+|AC|^2=|AB|^2

**End of the proof.**

** **

---

Source: [https://yvision.kz/post/pythagorean-theorem-a-2-b-2-c-2-621205](https://yvision.kz/post/pythagorean-theorem-a-2-b-2-c-2-621205)