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If in triangle ABC, b+c/11=c+a/12=a+b/13 prove that cosA:cosB:cosC=7:19:25.
Let ,b+c/11=c+a/12=a+b/13 = k b+c = 11k c+a = 12k a+b = 13k Add a +b +c = 18k a = 7k b= 6k c = 5kcos A = (6k) 2 + (5k) 2 - (7k) 2 )/2(6k)(5k) = 1/5 = 7/35cos B = (7k) 2 + (5k) 2 - (6k) 2 )/2(7k)(5k) = 19/35cos C = (6k) 2 + (7k) 2 - (5k) 2 )/2(6k)(7k) = 5/7 = 25/35cosA/7=cosB/19=cosC/25
Let ,b+c/11=c+a/12=a+b/13 = k b+c = 11k c+a = 12k a+b = 13k Add a +b +c = 18k a = 7k b= 6k c = 5k
cos A = (6k) 2 + (5k) 2 - (7k) 2 )/2(6k)(5k) = 1/5 = 7/35
cos B = (7k) 2 + (5k) 2 - (6k) 2 )/2(7k)(5k) = 19/35
cos C = (6k) 2 + (7k) 2 - (5k) 2 )/2(6k)(7k) = 5/7 = 25/35
cosA/7=cosB/19=cosC/25
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