President Garfield's proof of the Pythagorean Theorem - May 8-12, 1995 Anthony and Flea walked into geometry class and dropped a book on the table. "Check this out," said Anthony. "Flea and I are working on some extra credit for history class."
"Oh, right," said Chad. "And you want me to figure out the answer for you, just like last time."
"Hey!" said Flea. "That's water under the bridge. We've moved up to higher ground since then--this problem is interesting!"
"Okay," relented Chad. "So do you want me to hint at the answer when I figure it out, or just give it away?"
"Just give us a hand, will you?" asked Anthony, as he opened the book. "We were talking in history today about President James Garfield, and the book says that in addition to doing lots of Presidential things, he figured out a proof of the Pythagorean Theorem. Trying to look like he was paying attention, our friend Flea here asked what the proof was."
"And so you guys were given that very question as an extra credit problem, right?" asked Chad.
"Yeah," said Anthony, pointing to a diagram in the book. "Here's how the picture is constructed:
Construct a right triangle. Label it ABC as shown. Using point B as a center, rotate side c and point A by 90 degrees (to get E). Connect points A and E and construct a line through point E parallel to side b. Extend side CB and label the point of intersection with the line through E as D.
^
D| E
<--o------------o------------>
| /*
| /
| / *
| /
| / *
| /
| / *
| /
| / *
| /
| / *
|/
B o_ *
| * _
| * _ c *
a | * _
| * _ *
o__________________* o
C b A
So from all of this, we have to prove that a^2 + b^2 = c^2. Here's what we've done so far. We've seen other proofs of this theorem that use areas, and seeing that all those squared's mean area, that makes sense. But we weren't really sure where to go from there...I guess we just start figuring out all the areas we can find using a, b, and c?"
"Yeah," said Chad, "and don't forget that trapezoid!"
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