In 4ABC, altitude AD = 24, inradius of 4ABD = 3 and inradius of 4ACD = 6. Find BC.

Dear ajinkya

tanB = AD/BD  = 24/BD

tanB/2  = r1/(BD-r1)

tanB = 2tanB/2 /( 1 - tan²B/2)

  24/BD     = 2 {r1/(BD-r1)}/ 1- { r1/(BD-r1)}²

 BD = (24r1 -r1² )/(12-r1)

 r1 =3

 BD = 7

similerly

 CD = 18

BC = 7+18 = 25

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