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 The Law of Sines (Posted on 2005-12-13)
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 | 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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