Friday, January 19, 2007

Triangularly Speaking. Topic: Algebra/Geometry. Level: AIME.

Problem: Suppose the roots of the cubic polynomial $ P(x) = x^3-2rx^2+sx+t $ are the sides of a triangle (assume they are all positive reals and satisfy the triangle inequality). What is the area of the triangle in terms of the coefficients?

Solution: Well, let's think about the formulas for the area of a triangle given its side lengths. The only one that comes to mind that uses only side lengths is Heron's formula. Why don't we try that?

$ A = \sqrt{s(s-a)(s-b)(s-c)} $,

where $ a, b, c $ are the side lengths and $ s = \frac{a+b+c}{2} $. From $ P(x) $, by Vieta's, we know that $ r_1+r_2+r_3 = 2r $, where $ r_i $ are the roots of the polynomial. So $ s $ in this case is $ r $. How do we get the rest, though? We could expand and use Newton sums, but that sounds like a pain. Note that

$ (s-a)(s-b)(s-c) $ looks a lot like $ (x-a)(x-b)(x-c) $.

But wait, that's just a polynomial with roots $ a, b, c $! So if we write $ P(x) = (x-r_1)(x-r_2)(x-r_3) $, we have $ (s-a)(s-b)(s-c) = (s-r_1)(s-r_2)(s-r_3) = P(s) = P(r) $. Thus we can say that the area of the triangle is

$ \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{rP(r)} $,

which can clearly be evaluated in terms of the coefficients. QED.


Comment: This is probably one of the neatest tricks that Vieta's can help you with, as well as a cool link between algebra and geometry. Before I thought of this problem, I had never realized the now-clear relationship between Heron's formula and a cubic polynomial with sides as the roots. A nice discovery.


Practice Problem: (2007 Mock AMC 12 - #24) The lengths of the sides of a triangle are the roots of the equation $ 7x^3+35x = 28x^2+13 $. If the square of the area of the triangle is $ \displaystyle \frac{p}{q} $, where $ p $ and $ q $ are relatively prime positive integers, what is $ p+q $?


  1. Wow, that is a really nice problem.

    Let the roots be a,b,c. We have p(x):=7x^3 - 28x^2 + 35x - 13 = 0. a+b+c=1/4 so s=1/8. (s-a)(s-b)(s-c) = (2-a)(2-b)(2-c) = p(2) = 1. The square of the area is (s)(s-a)(s-b)(s-c) = 1/8 and 1+8 = 9.

  2. hey that's really cool, mainly cause i get it but also because... well i get it

  3. @Adeel: Lol, funny how you did it wrong and got the right answer... a+b+c = 4, so s = 2... For some reason you plugged s = 2 in even though you said s = 1/8. In any case, at the end it should be s(s-a)(s-b)(s-c) = 2P(2)/7 because the leading coefficient was 7. So the answer is 2/7 => 9.

  4. Lol... yeah, I accidentally did a+b+c=4 => s=2 at first, but then I fixed it, but forgot to fix everything else...