
Re: triangular numbers and dets
Posted:
Apr 25, 2012 5:05 PM


By the way I just tried the email you noted and it immediately bounced back, saying "undeliverable".
Anyway some remarks... ===================================
If the angle between the 7 and 33 sides is 60 then the third side satisfies
c^2 = 7^2 +33^2  2*7*33*cos(60), which since cos(60) = 1/2 gives
c^2 = 7^2  7*33 + 33^2 so that c = 37.
[I didn't see the relevance of the equation to triangles...]
The area formula for a triangle of sides a,b with angle theta included is
A = (1/2)*a*b*sin(theta).
So here the triangle has area (1/2)*7*33*sin(60) = (231/4)*sqrt(3).
So it's really only a coincidence it comes out nearly an integer...
In fact with integer sides for a,b and integer c with angle 60 between
the shorter sides, the area will always be a rational multiple of sqrt(3).
I've heard of the topic of finding triangles with integer sides and area,
I think they're called "Heronian triangles"...

