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In convex quadrilateral A B C D ,A B=a ,...

In convex quadrilateral `A B C D ,A B=a ,B C=b ,C D=c ,D A=d` . This quadrilateral is such that a circle can be inscribed in it and a circle can also be circumscribed about it. Prove that `tan^2A/2=(b c)/(a d)dot`

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Since a circle can be inscribed in the quadrilateral, we have `a + c = b + d`.

Further, it is also concyclic, thus `angle A + angle C = pi`
In `Delta BAD, BD^(2) = a^(2) + d^(2) - 2 ad cos A`
In `Delta BCD, BD^(2) = b^(2) + c^(2) - 2 bc cos C`
`= b^(2) + c^(2) + 2bc cos A`
`:. 2 cos A (bc + ad) = a^(2) + d^(2) - b^(2) -c^(2)`
or `2 cos A = (a^(2) + d^(2) - b^(2) - c^(2))/(bc + ad)`
Since `a + c = b + d, a - d = b - c`
`rArr a^(2) + d^(2) - 2ad = b^(2) + c^(2) - 2bc`
`rArr a^(2) + d^(2) - b^(2) - c^(2) = 2 (ad - bc)`
`:. cos A = (ad - bc)/(bc + ad) = (1 - tan^(2). (A)/(2))/(1 + tan^(2).(A)/(2))`
`rArr tan^(2). (A)/(2) = (bc)/(ad)`
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