Math Is Fun Forum

Hello again anonimnystefy, and hello PatternMan.

I've just started to learn Latex so perhaps I could practise it by adding here the formula for the number of ways that n numbers can be combined to illustrate the Associative Law:

(The brackets denote a binomial coefficient).

A couple of examples:

With n=3 numbers (1,2,3) there are 2 ways to associate them:

1. (1 + 2) + 3
2.  1 + (2 + 3)

With n=4 numbers (1,2,3,4) there are 5 ways to associate them:

1. 1 x (2 x (3 x 4))
2. 1 x ((2 x 3) x 4)
3. (1 x 2) x (3 x 4)
4. ((1 x 2) x 3) x 4
5. (1 x (2 x 3)) x 4

The above proof works for all triangles ABC, but it needs a small clarification when angle ACB > 120 degrees.

When angle ACB = 120 degrees, angles ACX and YCB become 180 degrees each, and the triangles ACX and YCB become trivial - they "collapse" into straight line segments of equal length. The proof still holds, and AX = YB (they are the line segments).

When angle ACB > 120 degrees, triangles ACX and YCB both now lie entirely outside  triangle ABC.
It is still true that angle ACX = angle YCB = (angle ACB + 60) but now we are talking about the EXTERNAL angles ACX and YCB (measuring them outside their triangles, the "long way round").

Therefore angle ACX = angle YCB (whether measured internally or externally) and the proof holds as before, with triangles ACX and YCB congruent.

Apologies for the lack of a diagram; I've only just joined and haven't learnt how to add diagrams yet. It should be easy to follow the above if you sketch it out.