SOLUTION: Hi am slightly stuck on this question in geometry Given a triangle ABC prove that: abc=4∆R where ∆ is the area of the triangle and R is the radius of the circumci

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Question 436544: Hi am slightly stuck on this question in geometry
Given a triangle ABC prove that:
abc=4∆R where ∆ is the area of the triangle and R is the radius of the circumcircle of the triangle

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!


Assume a,b,c are the side lengths opposite vertices A,B,C respectively. From the extended law of sines,

a%2Fsin%28A%29+=+b%2Fsin%28B%29+=+c%2Fsin%28C%29+=+2R; this implies sin%28A%29+=+a%2F2R.

Note that the area of the triangle is equal to DELTA+=+bc%2Asin%28A%29%2F2. We can substitute sin%28A%29 to obtain

DELTA+=+bc%28a%2F2R%29%2F2

DELTA+=+abc%2F4R

abc+=+4R%2ADELTA as desired.