Did anything stand out? maybe the m^2+n^2 = f^2 bit? That is Pythagoras’s theorem. That? This is what it’s like:
Pythagoras’s theorem states that, if the triangle is a right-angled triangle, a^2 + b^2 = c^2. Sure, this is one proof:
But for a better proof, it’s going to be awesome. First, draw lines, which will now be:
Now, since BF = BA and BC = BD, triangles BFC and ABD are congruent. Also, since A, K and L are collinear (all three points can form a straight line), and AL is parallel to BD, triangle ABD is half of rectangle BDLK. In the same way, BAGF is twice of FBC.
Since ABD and FBC are congruent, BDLK must be equal in area to BAGF. That is AB^2. WHEW! Also, draw lines AE and IB to find that KCEL = AHIC. Also known as AC^2!
Because the square BC^2 is made up of BDLK (aka. AB^2) and KCEL (aka. AC^2), we can now say that AB^2 + AC^2 = BC^2.
And that, everyone, is the Pythagorean Theorem (aka. Pythagoras’s Theorem) and I hope you enjoyed proving it.