### Degree Ceremony

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

### Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

### After Thought

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

# 8 Methods for Three by One

##### Stage: 4 and 5 Challenge Level:
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