# proof of Pythagorean theorem

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

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

 $\displaystyle\angle BAT$ $\displaystyle=$ $\displaystyle\angle ACT$ $\displaystyle\angle TAC$ $\displaystyle=$ $\displaystyle\angle CBA$

and therefore we have the following triangle similarities:

 $\triangle ABC\sim\triangle TBA\sim\triangle TAC.$

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

 $AB^{2}+AC^{2}=BC(BT+TC)=BC\cdot BC=BC^{2}$

which concludes the proof.

Title proof of Pythagorean theorem ProofOfPythagoreanTheorem1 2013-03-22 12:48:39 2013-03-22 12:48:39 drini (3) drini (3) 8 drini (3) Proof msc 51-00