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Olympiad workbookIOQMProperties of Triangles

Let ABC be an isosceles triangle with AB = BC . A trisector of B meets AC at D . If AB , AC and B D are integers and A B-B D=3 , find A C .

Correct answer

26

Step-by-step solution

aligned & I₁ ( 2 + ) = I₃ ( 2 -3 ) & put I₃=I₁-3 & I₁= 3 4 cosec ^2 & If cosec ^2 =36 & i.e. cosec ^2 =6 aligned aligned & i.e. = 1 6 & I₁= 3 4 36=27 & I₃=24 & I₁ ( 2 - ) = D C 4 & I₁ ( 2 + ) = A D 2 & I₂= I₁ 4 + I₁ 2 & I₂=I₁ 2 3 aligned aligned & 3 4 cosec ^2 2 3 & 3 2 ( 3 -4 ) I ₂ & = 3 2 (3 6- 4 6 ) & = 3 2 (18- 2 3 ) & = 3 2 ( 54-2 3 ) & = 3 2 52 3 I ₂ & =26 aligned

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