Show, without invoking the Pythagorean theorem, that the $3-4-5$ triangle is right The ancient Egyptians knew the $3-4-5$ triangle was a right triangle, but they did not possess the Pythagorean theorem or any equivalent theory.  Can it be shown that the $3-4-5$ triangle is a right triangle without using the Pythagorean theorem or any ideas related to it?  
This problem was shown to me by a fellow peer tutor a while ago.  I gave it some thought initially, but then I gave up trying when I realized it might not be doable.  What do you think?   

Take a stick length $3$ and another length $4$. Place these at right-angles to each other. Create a stick to measure the diagonal. Demonstrate that this stick plus the $3$-stick is the same length as two $4$-sticks. (You can create a $3$-stick from a $4$-stick by bisecting a $4$-stick, appending, and bisecting again).
