Algebra.Com's Answer #703720 by ikleyn(52914)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "In triangle ABC, the value of acotA+bcotB+ccotC is?
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document.write( "a)R+r
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document.write( "b)(R+r)/R
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document.write( "c)2(R+r)
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document.write( "d)3(R+r)
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document.write( "Also explain how?
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document.write( "It requires two ideas.\r
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document.write( "1. a*cot(A) + b*cot(B) + c*cot(C) =  =  . (1)\r\n" );
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document.write( " Next,  = 2R,  = 2R and  = 2R, where R is the radius of the circumscribed circle around the triangle,\r\n" );
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document.write( " according to the Sine Law theorem (see the lessons Law of sines and Law of sines - the Geometric Proof in this site).\r\n" );
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document.write( " Therefore, the line (1) can be continued in this way\r\n" );
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document.write( " a*cot(A) + b*cot(B) + c*cot(C) = 2R*(sin(A) + sin(B) + sin(C)). (2)\r\n" );
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document.write( " It is the first idea, and it allows us to reduce the problem to calculation of sin(A) + sin(B) + sin(C).\r\n" );
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document.write( "2. The second idea is THIS:\r\n" );
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document.write( " For any triangle with angles A, B and C\r\n" );
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document.write( " sin(A) + sin(B) + sin(C) =  , (3) \r\n" );
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document.write( " where r is the radius of the inscribed circle, while R is the radius of the circumscribed circle about the triangle.\r\n" );
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document.write( " Deriving formula (3) requires some technique, but it is known proof, which you can find at this reference\r\n" );
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document.write( " https://math.stackexchange.com/questions/734395/how-to-prove-that-fracrr1-cos-a-cos-b-cos-c\r\n" );
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document.write( "3. Finally, a*cot(A) + b*cot(B) + c*cot(C) = 2R*(r/R + 1)}}} = 2*(R+r).\r\n" );
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document.write( "Solved.\r
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