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GeometryQuestion and Answers: Page 25
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$$\mathrm{Resolution}\:\mathrm{du}\:\mathrm{probldme}\:\mathrm{pose}\:\mathrm{par}\: \\ $$$$\mathrm{sonukgindia}\:\left(\mathrm{16}.\mathrm{7}.\mathrm{2023}\right) \\ $$$$\mathrm{voir}\:\:\mathrm{Q194819} \\ $$$$\bigtriangleup\boldsymbol{\mathrm{ABC}}\:\:\boldsymbol{\mathrm{AM}}=\boldsymbol{\mathrm{AN}}=\boldsymbol{\mathrm{AD}}\mathrm{cos}\:\frac{\boldsymbol{\alpha}}{\mathrm{2}} \\ $$$$\begin{cases}{\boldsymbol{\mathrm{AC}}=\boldsymbol{\mathrm{AM}}+\boldsymbol{\mathrm{MC}}=\mathrm{17}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}\right)}\\{\boldsymbol{\mathrm{AB}}=\boldsymbol{\mathrm{AN}}+\boldsymbol{\mathrm{NB}}\:\:=\mathrm{18}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\right)}\end{cases} \\ $$$$\:\:\boldsymbol{\mathrm{AB}}−\boldsymbol{\mathrm{AC}}=\mathrm{1}=\boldsymbol{\mathrm{NB}}−\boldsymbol{\mathrm{MC}}\:\:\:\left(\mathrm{3}\right) \\ $$$$\bigtriangleup\boldsymbol{\mathrm{CDE}}\:\:\:\mathrm{cos}\:\frac{\boldsymbol{\mathrm{C}}}{\mathrm{2}}=\frac{\boldsymbol{\mathrm{CM}}}{\boldsymbol{\mathrm{CD}}}=\frac{\boldsymbol{\mathrm{CE}}}{\boldsymbol{\mathrm{CD}}}\:\Rightarrow\boldsymbol{\mathrm{CM}}=\boldsymbol{\mathrm{CE}} \\ $$$$\bigtriangleup\boldsymbol{\mathrm{BDE}}\:\:\:\mathrm{cos}\:\frac{\boldsymbol{\mathrm{B}}}{\mathrm{2}}=\frac{\boldsymbol{\mathrm{BE}}}{\boldsymbol{\mathrm{BD}}}=\frac{\boldsymbol{\mathrm{BN}}}{\boldsymbol{\mathrm{BD}}}\Rightarrow\boldsymbol{\mathrm{BN}}=\boldsymbol{\mathrm{BE}} \\ $$$$\Rightarrow\boldsymbol{\mathrm{BE}}−\boldsymbol{\mathrm{CE}}=\mathrm{1} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{BC}}=\boldsymbol{\mathrm{BE}}+\boldsymbol{\mathrm{CE}}=\mathrm{2}\boldsymbol{\mathrm{BE}}−\mathrm{1} \\ $$$$\:\bigtriangleup\boldsymbol{\mathrm{BEF}}\:\:\:\boldsymbol{\mathrm{BF}}=\mathrm{9}\:\:\:\:\mathrm{cos}\:\boldsymbol{\mathrm{B}}=\frac{\boldsymbol{\mathrm{BE}}}{\mathrm{9}}\:\: \\ $$$$\:\:\boldsymbol{\mathrm{BE}}=\mathrm{9cos}\:\boldsymbol{\mathrm{B}}\:\:\:\:\Rightarrow\boldsymbol{\mathrm{BC}}=\mathrm{18cos}\:\boldsymbol{\mathrm{B}}−\mathrm{1} \\ $$$$\:\:\boldsymbol{\mathrm{Posons}}\:\:\boldsymbol{\mathrm{BC}}=\boldsymbol{\mathrm{x}}\:\:\:\:\boldsymbol{\mathrm{x}}=\mathrm{18cos}\:\boldsymbol{\mathrm{B}}−\mathrm{1} \\ $$$$ \\ $$$$\:\:\boldsymbol{\mathrm{d}}\:\boldsymbol{\mathrm{apres}}\:\boldsymbol{\mathrm{triangle}}\:\:\boldsymbol{\mathrm{ABC}} \\ $$$$\:\boldsymbol{\mathrm{AC}}^{\mathrm{2}} =\boldsymbol{\mathrm{AB}}^{\mathrm{2}} +\boldsymbol{\mathrm{BC}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{\mathrm{AB}}.\boldsymbol{\mathrm{BC}}\mathrm{cos}\:\boldsymbol{\mathrm{B}} \\ $$$$\:\:\:\: \\ $$$$\Rightarrow\mathrm{17}^{\mathrm{2}} =\mathrm{18}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{36}\left(\frac{\boldsymbol{\mathrm{x}}+\mathrm{1}}{\mathrm{18}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{\mathrm{x}}−\mathrm{35}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{alors}}\:\:\:\:\:\:\boldsymbol{\mathrm{x}}=\mathrm{5} \\ $$
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$$\:\:\:\:\:\:\underbrace{\:} \\ $$
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$$\mathrm{soit}\:\:\:\boldsymbol{\mathrm{A}}\left(\mathrm{2},\mathrm{1}\right)\:\:\:\:\:\boldsymbol{\mathrm{B}}\left(\mathrm{3},\mathrm{2}\right)\:\:\:\:\:\boldsymbol{\mathrm{C}}\left(\mathrm{4},\mathrm{3}\right)\:\:\mathrm{points}\:\mathrm{du}\: \\ $$$$\:\:\mathrm{plan}\left(\:\mathrm{ox},\mathrm{oy}\right)\:\: \\ $$$$ \\ $$$$\left.\:\mathrm{1}\right)\mathrm{Determiner}\:\:\mathrm{l}\:'\:\mathrm{equation}\:\mathrm{du}\:\mathrm{cercle}\:\mathrm{qui}\: \\ $$$$\:\:\:\:\mathrm{passe}\:\mathrm{par}\:\boldsymbol{\mathrm{A}};\:\boldsymbol{\mathrm{B}};\:\boldsymbol{\mathrm{C}}\:\:\:? \\ $$$$\left.\:\:\mathrm{2}\right)\:\mathrm{points}\:\mathrm{d}\:\mathrm{intersection}\:\mathrm{du}\:\mathrm{cercle}\:\mathrm{avec} \\ $$$$\:\:\:\:\:\mathrm{l}\:\mathrm{axe}\left(\mathrm{ox},\mathrm{oy}\right)? \\ $$
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$$\mathrm{Determiner}\:\mathrm{le}\:\mathrm{rayon}\:\boldsymbol{\mathrm{r}} \\ $$
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