` Delta ABC` में `tan . A/2 = 5/6 ` और ` tan . B/2 = 20/37`
तो सिद्ध कीजिए कि a ,b,c समांतर श्रेणी में है ।

Explanation

` tan.C/2 = tan {(180^(@) - (A+B))/2}`
` = tan { 90^(@) - (A/2 +B/2 )}`
` = cot (A/2 +B/2)`
` = 1/(tan. (A/2+B/2))= (1- tan. A/2 tan.B/2)/(tan.A/2 +tan.B/2)`
` = (1- 5/6 xx 20/37)/(5/6 +20/37)= (6 xx 37 - 5 xx 20 )/(5 xx 37 + 6 xx 20)`
` (222-100)/(185+120)= 122/305 = 2/5`
` sin A = ( 2 tan. A/2 )/(1+ tan^(2) A/2 )= (2xx 5/6)/(1+(5/6)^(2))`
` = 5/3 xx 36/ 61 = 60/61 `
` sin B = (2 tan. B/2 )/(1+ tan^(2). B/2 )= (2 xx 20/37)/(1+(20/37)^(2))`
` = 40/ 37 xx 1369/1769 = 1480/1769`
और `sin C = (2 tan .C/2)/(1+tan^(2).C/2 )`
` = (2 xx2/5)/(1+(2/5)^(2))= 4/5 xx 25/29= 20/29`
` :. a :b:c = sin A : sin B : sin C `
` = 60/ 61 : 1480/1769: 20/29`
` = 60 xx 29 : 1480 : 20 xx 61 `
` = 1740 : 1480 : 1220`
` = 87 : 74 : 61 `
माना ` a = 87 k, b= 74k, c= 61 k`
अब ` b-a = 74 k - 87 k = - 13 k `
` c- b = 61 k - 74 k = - 13k `
` :' b- a = c - b `
` :' a, b, c ` समांतर श्रेढ़ी में है ।