Saturday, May 10, 2014

Triangle theorems

Most of the theorems about triangles that I read about date from antiquity.  For instance, Euclid's Elements contains many propositions such as
  • The sum of the length of two sides of a triangle is larger than the length of the third side.
  • An isosceles triangle has the two angles at the base equal to each other.
  • Pythagoras Theorem
An interesting theorem is Heron's formula (probably known in Archimedes' time) for the area $A$ of a triangle with sides of lengths a, b,c:
\[ A = \sqrt{s(s-a)(s-b)(s-c)}\]
where $s$ is the semiperimeter defined as $s = \frac{1}{2}(a+b+c)$.  This theorem allows you to calculate the area of a triangle using solely the lengths of the 3 sides.  A similar theorem was discovered by the Chinese independently from the Greeks.

Of more recent vintage is Routh's theorem, published in 1896, that describes the ratio between a triangle and another triangle generated via the intersection of 3 cevians.

I was pleasantly surprised to hear about 2 new theorems about triangles that have been proved only relatively recently.

  1. Conway's little theorem (1976): A triangle is equilateral if and only if the ratio between the length of any 2 sides is rational and the ratio between any 2 angles is rational.
  2. Grieser and Maronna 2013: Up to congruence, a triangle is determined uniquely by its area, perimeter and the sum of the reciprocals of its angles. 
  • J. Conway, "A Characterization of the Equilateral Triangles and Some Consequences," The Mathematical Intelligencer, 20 March 2014.
  • D. Grieser and S. Maronna, "Hearing the Shape of a Triangle," Notices of AMS, vol. 60, no. 11, pp. 1440-1447, 2013. 

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