[Today’s run: 3.4 miles with wife and dog]
I was speaking to my nephew the other day and he mentioned the Pythagorean Theorem: That for a right triangle the sum of the squares of the lengths of the two sides adjacent to the 90 degree corner (the short sides) is equal to the sum of the length of the hypotenuse (the long side): Squared A + Squared B = Squared C.
I thought to myself, “I wonder if I can prove the Pythagorean Thorem.” I understand there are a whole bunch of proofs for it. I had just read about some guy in the Civil War time (I think it was) who came up with a “graphical proof”. (President Garfield… I googled that.)
I’m not normally a mathematician. That I can easily prove!
I’ve spent a few hours over the last week or so trying to come up with a proof. No joy yet. But I haven’t given up. Eventually I will either prove it or just give up and google it.
I was working on the idea of extending the triangle to encompass the square made by squaring side C. I noticed that C * H = A * B where H is the perpendicular height of the ABC triangle from line C to the 90 degree vertex.
I came up with equations for the larger triangle based on C, H, A and B. I thought I could make two equations of area for the larger triangle, T: one based on adding up the areas of square C and the three smaller triangles, and one based on bigT(A) and bigT(B); then solve for C-squared in terms of A and B. But I have not been able to get completely through. Somehow my algebraic terms all cancel out and leave C-squared = C-squared. Hardly what we want.
I think I could finish that line of advance if I assumed that the theorem is true. But that doesn’t seem like a fair way to prove something.
So I am looking for another angle of attack (so to speak).
I just thought of another equation for the big triangle T. Now I have three equations for the area of big triangle T.
Don’t tell me! I’m going to figure it out.