proof of Pythagorean theorem

Let $A B C$ be a right triangle with hypotenuse $B C$ . Draw the height $A T$ .

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.