Thus starting with the triangle 1 we add three more in the way suggested in proof #7: similar and similarly described triangles 2, 3, and 4.
Deriving a couple of ratios as was done in proof #6 we arrive at the side lengths as depicted on the diagram.
The Pythagorean configuration is known under many names, the Bride's Chair being probably the most popular.
And, since ABC is similar to ABD, therefore, as CB is to BA so is AB to BD [VI. And, since three straight lines are proportional, as the first is to the third, so is the figure on the first to the similar and similarly described figure on the second [VI.19]. I got a real appreciation of this proof only after reading the book by Polya I mentioned above.
Therefore, as CB is to BD, so is the figure on CB to the similar and similarly described figure on BA. I hope that a Java applet will help you get to the bottom of this remarkable proof.
For the same reason also, as BC is to CD, so is the figure on BC to that on CA; so that, in addition, as BC is to BD, DC, so is the figure on BC to the similar and similarly described figures on BA, AC. Note that the statement actually proven is much more general than the theorem as it's generally known.
But BC is equal to BD, DC; therefore the figure on BC is also equal to the similar and similarly described figures on BA, AC. Playing with the applet that demonstrates the Euclid's proof (#7), I have discovered another one which, although ugly, serves the purpose nonetheless.
The construction did not start with a triangle but now we draw two of them, both with sides a and b and hypotenuse c.
Note that the segment common to the two squares has been removed.There is nothing much one can add to the two pictures.(My sincere thanks go to Monty Phister for the kind permission to use the graphics.) There is an interactive simulation to toy with.In addition, it highlights the relation of the latter to proof #1.(II.5) which is a recommended reading to students and teachers of Mathematics.For example, in the diagram three points F, G, H located on the circle form another right triangle with the altitude FK of length a.