There is a triangle ABC on a euclidean plane. Like every other triangle on the plane, it follows the law of sines, that is, BC/sin(A) = AC/sin(B) = AB/sin(C).
So we know that these three numbers are equal to one another, but most people don't know that they are also equal to the length of a special line segment. What is the significance of this length, and can you prove it?
For triangle ABC let M be the circumcenter, R the circumradius, and
x = <MBC = <MCB
y = <MCA = <MAC
z = <MAB = <MBA
Then
BC 2*R*cos(x) 2*R*cos(x) 2*R*cos(x)
 =  =  =  = 2*R
sin(A) sin(y+z) sin(90x) cos(x)
Therefore, the length is the length of the diameter of the circumcircle.

Posted by Bractals
on 20051213 13:00:18 