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The Law of Sines (Posted on 2005-12-13) Difficulty: 3 of 5
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?

See The Solution Submitted by Tristan    
Rating: 3.0000 (2 votes)

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Solution Solution | Comment 3 of 4 |
 
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(90-x)       cos(x)
Therefore, the length is the length of the diameter of the circumcircle.
 

  Posted by Bractals on 2005-12-13 13:00:18
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