Look at the first and third figures and try to answer this. How can one SIMPLY convince me that right triangle with the legs of 3 and 4 units must have exactly 5 units long hypotenuse. Exactly? Why is that? I don't get it. I don't feel it. Just like many people don't. I don't know algebra, I don't know Pythagorean theorem, I don't understand symbolics and formulas. And mathematical proofs don't make sense to me, as well as, for many students, I think. You know such people and students - they really don't get it. And it is not surprising. Common sense is what they need, and I ask for.

Or, how could ancient Babylonians, far before Pythagorean times, know about these integer sided right triangles. They knew a dozen of them, the classical Babylonian triangle (3,4,5), or (5,12,13), and more... How could they come up with such discoveries not knowing algebra or PyThm?

Measure it? Accurately? Whatever accuracy is, it still can't give me 5 EXACTLY.

A few days ago I was talking to a carpenter who came up with this very same question? He did use 3,4,5 triangular rope to construct right angle in the field. It worked. But he is still curious why this trick works. I told him and he was so happy. He got it. It made sense for him. He doesn't know abstract things, just numbers and simple drawings.

What I am going to do is to simply repeat the ideas re algebraic proof of PyThm you discussed (Alg.2), but reduced to concrete numbers, following trivial and naive steps, actually counting. As a result, I will discover my own mini-theorem, provide a mini-proof, and get the right answer, which will be truly understandable to everyone!

Make this SIMPLE DESIGN (middle) on grid paper: Square-in-square turned. You recognize it! What's the area of the big square? 7 by 7 which gives 49. Just count the number of unit squares inside. What's the area of all four corner triangles? Two of them (opposite ones) slide to form a rectangle 3 by 4, i.e 12. Just, count the squares in that rectangle to find 12. All 4 triangles have area twice, i.e 24 units.

What is left for the inner red square? 49-24 = 25. Wow, 25 exactly! A square with area 25. What is its side which if multiplied by itself gives 25? Or draw separately a square on a grid paper along the gridlines, with area of 25 small unit squares. You see that the side has to be exactly 5 units! Wow.

Not only we almost discovered our own Pythagorean-like theorem, we almost proved it (partially)! But in fact we avoided (bypassed) Pythagorean theorem! We didn't get the theorem itself, but we applied its power in practice. It's simple, it works, it makes sense. It gives a better insight on that sophisticated algebraic proof of PyThm. Everyone can understand this.

Now try 5 by 12 right triangle to find 13! Big square has area 17x17=289. Two corner triangles give 5x12=60, and four of them give twice that, 120. What is left for the inner square? 289-120=169. Is there an integer whose square is 169? Check it out.

Intuitive History. This is how the ancients before Pythagoras could easily fimd the Pyth. triples!

Try other lengths between any two lattice points on grid paper. You will come up with square roots of some Area-numbers, which are not always integers. For instance, try 1 by 2. Create the simple square-in-square diagram. The big square has area (1+2)x(1+2) = 3x3 = 9. Two corner triangles give a 1 by 2 rectangle, i.e. area 2, and all four triangles have twice that area: 4 unit squares. What's left for the inner square is 9 - 4 = 5. Okay, a square has area 5. What's its side. No integer fits. We define it as square root of 5. We just define and calculate using trial and error (use calculator) to find it approximately 2.23...whose square is 5.

Finding distances between lattice points on grid paper can be done nicely using GeoBoard in Geometry course, before even getting to Pythagorean theorem.

Mamikon