proof of Pythagorean theorem

Let $ABC$ be a right triangle with hypotenuse $BC$. Draw the height $AT$.

Using the right angles $\mathrm{\angle }BAC$ and $\mathrm{\angle }ATB$ and the fact that the sum of angles on any triangle is ${180}^{\circ }$, it can be shown that

$\mathrm{\angle }BAT$ $=$ $\mathrm{\angle }ACT$

$\mathrm{\angle }TAC$ $=$ $\mathrm{\angle }CBA$

and therefore we have the following triangle similarities:

$$\mathrm{△}ABC\sim \mathrm{△}TBA\sim \mathrm{△}TAC.$$

From those similarities, we have $\frac{AB}{BC}=\frac{TB}{BA}$ and thus $A{B}^2=BC\cdot TB$. Also $\frac{AC}{BC}=\frac{TC}{AC}$ and thus $A{C}^2=BC\cdot TC$. We have then

$$A{B}^2+A{C}^2=BC(BT+TC)=BC\cdot BC=B{C}^2$$

which concludes the proof.

Title	proof of Pythagorean theorem
Canonical name	ProofOfPythagoreanTheorem1
Date of creation	2013-03-22 12:48:39
Last modified on	2013-03-22 12:48:39
Owner	drini (3)
Last modified by	drini (3)
Numerical id	8
Author	drini (3)
Entry type	Proof
Classification	msc 51-00