Comparing and Contrasting
#1: This proof, Euclid's proof, fits in a lot of categories. First of all, triangles shift a lot, shearing and turning and rotating and squeezing into the half of their rectangles.
SIMILARITIES: 1.Both involve congruent triangles that are necessary to shift or prove congruent areas.
2.Both are geometric proofs (this is really obvious so we won't show it anymore).
3.Both involve squares on the right triangle's sides to show congruent triangles: a^2 + b^2 = c^2.
4.Both involve proving non-right triangles congruent to prove congruent area.
5.Both involve congruent angles and segments to prove congruent triangles.
2.Both are geometric proofs (this is really obvious so we won't show it anymore).
3.Both involve squares on the right triangle's sides to show congruent triangles: a^2 + b^2 = c^2.
4.Both involve proving non-right triangles congruent to prove congruent area.
5.Both involve congruent angles and segments to prove congruent triangles.
#36: In the first variation of this proof, it leaves triangles where they originally were. (The exact opposite of #1!) This proof involves proving lots of triangles congruent, using SAS and ASA and opposite int. angles and corresponding angles.
DIFFERENCES: In Proof #1, the triangles are shifted around a lot, but the areas are the same. Not so for #36. There, the triangles are left where they are. Also, in #1, only two groups of triangles are proved congruent, where in #36, four groups are proved congruent. In #36, 3 theorems are involved. In #1, only 1 (SAS) is used. Another thing: #36 involves scaling triangles down. #1: Nope.
Below: Similarity examples!