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'8 Methods for Three by One' printed from http://nrich.maths.org/
Consider this problem: for the following diagram, prove that $a+b=c$
See how you would go about solving it.
Now, this problem has been shown on NRICH before. When first shown on NRICH it was solved in 8 different ways by a pair of students Alex and Neil. When we again showed the problem, Sigi sent us a lovely new geometric proof.
Sigi suggests that it would be interesting to look at the different proof methods and think about which are mathematically independent of each other in that they use genuinely distinct mathematical ideas rather than the same ideas dressed up in different ways.
We agree with Sigi: Analyse each of the proof methods. How many genuninely distinct methods can be found? In what ways are they different from each other? Do you have a favourite proof? Do some proof methods seem to have potential for wider generalisation.
Sigi's proof is found here
The distinct proofs from Alex and Neil are:
Method 1: Tan Angle Sum Formula
Method 2: Sin Angle Sum Formula
Method 3: Cosine Rule
Method 4: Vector
Method 5: Matrices
Method 6: Pure Geometry
Method 7: Coordinate Geometry
Method 8: Complex Numbers