Math Is Fun Forum

This is a little trick I have found incredibly useful (especially in a past competition). The explanation is messy and bad, so please forgive me, I'm a noob at number theory too, so this is new for me. I'd be happy if someone could help clean up the explanation

If you have an equation of the form

With variables x,y and constants a,b,c,

It can be turned into the form:

Now introduce a new variable d such that

I find this very important as it allows you to be able to work out the possible numbers rather easily. The only real 'trick' which separates this from normal factorisation is the introduction of the new variable which is in ratio.

E.g Find all non-hypotenuse side lengths a,b of a right triangle, such that the hypotenuse of that triangle is equal to its area.

Now you can systematically work out the possible values of a and b. In this case they might not be rational but you get the idea.

Given

[previously proven], use corresponding results to prove the cosine rule.

Thanks

Write down two formulae for the area of Triangle ABC with sides a, b, and c, and deduce the sine rule from these two.

I'm assuming here the question is refering to the sine formulae for area and heron's formula, but it's really hard to derive because things get messy quickly with the heron's formula! Is there any simpler way to do things?

[don't mind with this question, a moderator can delete this post]