koli123able
Prove the followings
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koli123able
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Number 1 \[\frac{ \Delta }{ s-a }=s \tan \frac{ \alpha }{ 2 }\]
mathslover
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what does alpha , a , s , \(\Delta\) represent ?
koli123able
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\[\Delta=\sqrt{s(s-a)(s-b)(s-c)}\]
koli123able
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\[\tan \frac{ \alpha }{ 2 }=\sqrt{\frac{ (s-b)(s-c) }{ s(s-a) }}\]
mathslover
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right... got it now..
mathslover
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so now put these values :
\[\large{\frac{ \sqrt{s(s-a)(s-b)(s-c)}}{s-a} = \frac{\sqrt{s-a} \sqrt{s(s-b)(s-c)}}{s-a} }\]
koli123able
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prove it using either l.h.s or r.h.s
mathslover
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Yeah I am using LHS only
koli123able
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okay.. :)
mathslover
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well I can write :; \(\large{\frac{\sqrt{s-a}}{s-a} }\) as \(\frac{1}{\large{\sqrt{s-a}}}\)
mathslover
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therefore I get :
\[\large{\frac{\sqrt{s(s-b)(s-c)}}{\sqrt{s-a}}}\] = \[\large{\sqrt{\frac{(s-b)(s-c)s^2}{s(s-a)}}} \]
That is : \[\large{s\tan \frac{\alpha}{2}}\]
mathslover
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got it ?
koli123able
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can u prove it by multiplying and dividing trick i did'nt understand the roots u changed
koli123able
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like taking r.h.s and M & D it by \[\sqrt{s(s-a)}\]
mathslover
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see : |dw:1360917633731:dw|
koli123able
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NUMBER 2\[\frac{ s-a }{ \Delta }+\frac{ s-b }{ \Delta }+\frac{ s-c }{ \Delta }= \frac{ s }{ \Delta }\]
koli123able
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Using lhs
koli123able
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\[\frac{ s-a+s-b+s-c }{ \Delta }\]
koli123able
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what to do after this??
hartnn
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didn't you ask the first one earlier also? :O
also, for 2nd, we know that s is the semi-perimeter , hence,
\(s= (a+b+c)/2 ------> 2s = a+b+c\)
so, your numerator\( = s-a+s-b+s-c = 3s- (a+b+c) =.... ?\)
koli123able
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:D yes i did asked the first 1