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Question-22929




Question Number 22929 by selestian last updated on 24/Oct/17
Answered by ajfour last updated on 24/Oct/17
(A)  r(cot (A/2)+cot (B/2))=c  r(cot (B/2)+cot (C/2))=a  r(cot (C/2)+cot (A/2))=b  2b=a+c , so  2r(cot (C/2)+cot (A/2))=r(cot (A/2)+                                  +2cot (B/2)+cot (C/2))  ⇒   cot (C/2)+cot (A/2)=2cot (B/2)  For a triangle Σcot (A/2)=Πcot (A/2)  ⇒cot (A/2)cot (B/2)cot (C/2) −cot (B/2)                                            =2cot (B/2)  ⇒  cot (A/2)cot (C/2)=3  ⇒    cot (A/2)=3tan (A/2) .
$$\left({A}\right) \\ $$$${r}\left(\mathrm{cot}\:\frac{{A}}{\mathrm{2}}+\mathrm{cot}\:\frac{{B}}{\mathrm{2}}\right)={c} \\ $$$${r}\left(\mathrm{cot}\:\frac{{B}}{\mathrm{2}}+\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\right)={a} \\ $$$${r}\left(\mathrm{cot}\:\frac{{C}}{\mathrm{2}}+\mathrm{cot}\:\frac{{A}}{\mathrm{2}}\right)={b} \\ $$$$\mathrm{2}{b}={a}+{c}\:,\:{so} \\ $$$$\mathrm{2}{r}\left(\mathrm{cot}\:\frac{{C}}{\mathrm{2}}+\mathrm{cot}\:\frac{{A}}{\mathrm{2}}\right)={r}\left(\mathrm{cot}\:\frac{{A}}{\mathrm{2}}+\right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{2cot}\:\frac{{B}}{\mathrm{2}}+\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\right) \\ $$$$\Rightarrow\:\:\:\mathrm{cot}\:\frac{{C}}{\mathrm{2}}+\mathrm{cot}\:\frac{{A}}{\mathrm{2}}=\mathrm{2cot}\:\frac{{B}}{\mathrm{2}} \\ $$$${For}\:{a}\:{triangle}\:\Sigma\mathrm{cot}\:\frac{{A}}{\mathrm{2}}=\Pi\mathrm{cot}\:\frac{{A}}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{cot}\:\frac{{A}}{\mathrm{2}}\mathrm{cot}\:\frac{{B}}{\mathrm{2}}\mathrm{cot}\:\frac{{C}}{\mathrm{2}}\:−\mathrm{cot}\:\frac{{B}}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2cot}\:\frac{{B}}{\mathrm{2}} \\ $$$$\Rightarrow\:\:\mathrm{cot}\:\frac{{A}}{\mathrm{2}}\mathrm{cot}\:\frac{{C}}{\mathrm{2}}=\mathrm{3} \\ $$$$\Rightarrow\:\:\:\:\mathrm{cot}\:\frac{\boldsymbol{{A}}}{\mathrm{2}}=\mathrm{3tan}\:\frac{{A}}{\mathrm{2}}\:. \\ $$$$\:\:\:\: \\ $$$$ \\ $$
Commented by selestian last updated on 24/Oct/17
sir plz send me the solution plz plz
$${sir}\:{plz}\:{send}\:{me}\:{the}\:{solution}\:{plz}\:{plz} \\ $$$$ \\ $$
Commented by math solver last updated on 24/Oct/17
sir how you wrote the 1st 3 eq. ?  r(....)=a ..... ??
$${sir}\:{how}\:{you}\:{wrote}\:{the}\:\mathrm{1}{st}\:\mathrm{3}\:{eq}.\:? \\ $$$${r}\left(….\right)={a}\:…..\:?? \\ $$
Commented by ajfour last updated on 24/Oct/17
diagram ? which part especialy  do i explain more ?
$${diagram}\:?\:{which}\:{part}\:{especialy} \\ $$$${do}\:{i}\:{explain}\:{more}\:? \\ $$
Commented by selestian last updated on 24/Oct/17
thanks sir can you plZ xplain
$${thanks}\:{sir}\:{can}\:{you}\:{plZ}\:{xplain} \\ $$
Commented by ajfour last updated on 25/Oct/17
see Q.23009
$${see}\:{Q}.\mathrm{23009} \\ $$
Commented by selestian last updated on 25/Oct/17
plz xplAin from the line for a triangle
$${plz}\:{xplAin}\:{from}\:{the}\:{line}\:{for}\:{a}\:{triangle} \\ $$

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