exotic proof of Heron's formula

If a triangle has area A and sides of lengths x,y,z, then
16 A^2 = (x+y+z)(x+y-z)(x-y+z)(-x+y+z)
(Heron's formula).

Now here is today's homework from the Sortov Institute:

Let m and n be two real numbers, and define three more (real) numbers
by
a = sqrt(mm + nn)
b = sqrt(mm - 2mn + 2nn)
c = sqrt(2mm - 2mn + nn).
From the obvious identity
(a+b+c)(a+b-c)(a-b+c)(-a+b+c) = 4(mm - mn + nn)^2
deduce Heron's formula in a truly exotic way.

Hint: Diophantine approximation.

:D

.

Relevant Pages

  • Re: geometric proofs of incommensurable lengths
    ... If the two lengths had a common unit of measure, ... Suppose an isosceles right triangle has sides ... of length d and a hypothenuse of length h. ...
    (sci.math)
  • Re: Putnam Exam 2004 -- [*SPOILERS*]
    ... > expansion are non negative. ... the area and side lengths of a triangle satisfy ... >> Taylor series shows that the limit must equal this infinite sum. ...
    (sci.math)
  • fermat last
    ... Z of a right triangle. ... The lengths X',Y' and Z' are used to represent an arbitrary set of ... I have formed an infinite number of integer solutions, when n=2, using ...
    (sci.math)
  • Re: exotic proof of Herons formula
    ... If a triangle has area A and sides of lengths x,y,z, then ... Consider a tesselation of the unit torus by congruent (not necessarily ... The number of "lines" in the set having winding numbers is gcd. ...
    (sci.math)
  • Re: Trig Function seems odd.
    ... proportions. ... homework I'd probably be young enough to remember my maths classes. ... the same two sides in the large triangle. ... I think the real problem was that when I tried to be smart and ...
    (microsoft.public.excel.worksheet.functions)
Quantcast