Question Number 88188 by mr W last updated on 08/Apr/20
Commented by mr W last updated on 08/Apr/20
$${Given}:\:{triangle}\:{ABC}\:{with}\:{side}\:{lengthes} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a},\:{b},\:{c}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{P}\:{is}\:{midpoint}\:{of}\:{BC}. \\ $$$${Find}:\:\:\:{perimeter}\:{of}\:{inscribed}\:{triangle} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{PQR}\:{with}\:{the}\:{smallest}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{perimeter}. \\ $$
Commented by ajfour last updated on 09/Apr/20
Commented by ajfour last updated on 09/Apr/20
$${let}\:\angle{A}=\alpha\:,\:\angle{B}=\beta\:,\:\angle{C}=\gamma\:, \\ $$$$\angle{QPC}=\theta,\: \\ $$$${p}\left({slope}\:{of}\:{AB}'\right)=\mathrm{tan}\:\left(\pi−\mathrm{2}\gamma−\beta\right) \\ $$$$\mathrm{tan}\:\theta={m},\:\mathrm{tan}\:\beta={q},\:\:{A}\left({h},{k}\right) \\ $$$${AR}={BR}'\:=\:{l} \\ $$$${eq}.\:{of}\:{PQ}:\:\:\:{y}={m}\left({x}−{a}/\mathrm{2}\right) \\ $$$${eq}.\:{of}\:{A}\:'{B}\::\:\:\:{y}=−{qx} \\ $$$${eq}.\:{of}\:{AB}'\::\:\:{y}−{k}={p}\left({x}−{h}\right) \\ $$$$\underset{−} {{for}\:{R}'} \\ $$$${m}\left({x}−\frac{{a}}{\mathrm{2}}\right)=−{qx} \\ $$$${x}_{{R}'} =\frac{{m}\left({a}/\mathrm{2}\right)}{{q}+{m}}\:,\:\:{y}_{{R}'} =−\frac{{qm}\left({a}/\mathrm{2}\right)}{{q}+{m}} \\ $$$$\underset{−} {{for}\:{R}} \\ $$$${m}\left({x}−{a}/\mathrm{2}\right)={k}+{p}\left({x}−{h}\right) \\ $$$${x}_{{R}} =\frac{{k}−{ph}+{m}\left({a}/\mathrm{2}\right)}{{m}−{p}} \\ $$$${y}_{{R}} ={m}\left(\frac{{k}−{ph}+{p}\left({a}/\mathrm{2}\right)}{{m}−{p}}\right) \\ $$$${l}^{\mathrm{2}} =\left({BR}'\right)^{\mathrm{2}} =\left[\frac{{m}\left({a}/\mathrm{2}\right)}{{q}+{m}}\right]^{\mathrm{2}} \left(\mathrm{1}+{q}^{\mathrm{2}} \right) \\ $$$$\:\:\:=\left({AR}\right)^{\mathrm{2}} =\:\left(\frac{{k}−{mh}+{m}\left({a}/\mathrm{2}\right)}{{m}−{p}}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:+\left\{\frac{{m}\left[−{ph}+{p}\left({a}/\mathrm{2}\right)\right]+{pk}}{{m}−{p}}\right\}^{\mathrm{2}} \\ $$$${from}\:{above}\:{eq}.\:{we}\:{solve}\:{for}\:{m}. \\ $$$${Then}\:{min}\left(\:{perimeter}\:\bigtriangleup{PQR}\:\right) \\ $$$$\:\:\:\:\:=\:{R}'{R}\:. \\ $$
Commented by ajfour last updated on 09/Apr/20
$${couldn}'{t}\:{compress}\:{it}\:{Sir}! \\ $$
Commented by mr W last updated on 09/Apr/20
$${thank}\:{you}\:{sir}!\:{it}\:{is}\:{the}\:{geometric}\:{way} \\ $$$${which}\:{i}\:{prefer}.\:{i}'{ll}\:{try}\:{in}\:{a}\:{similar}\:{way}. \\ $$
Answered by mr W last updated on 09/Apr/20
Commented by mr W last updated on 09/Apr/20
$${let}\:\angle{A}=\alpha,\:\angle{B}=\beta,\:\angle{C}=\gamma \\ $$$${we}\:{construct}\:{the}\:{image}\:{of}\:{side}\:{BC} \\ $$$${about}\:{AB}\:{and}\:{about}\:{AC}.\:{P}^{\:} '\:{and}\:{P}\:'' \\ $$$${are}\:{the}\:{corresponding}\:{image}\:{of}\:{P}. \\ $$$${BP}\:'={BP}=\frac{{a}}{\mathrm{2}},\:{CP}\:''={CP}=\frac{{a}}{\mathrm{2}} \\ $$$${P}\:'{R}={PR},\:{QP}\:''={QP} \\ $$$${perimeter}\:{of}\:\Delta{PQR}={PQ}+{QR}+{PR} \\ $$$$={P}\:'{R}+{RQ}+{QP}\:'' \\ $$$${we}\:{see}\:{it}\:{is}\:{minimum},\:{if}\:{P}\:',{R},{Q},{P}\:'' \\ $$$${are}\:{collinear},\:{then} \\ $$$${perimeter}\:{of}\:\Delta{PQR}={P}\:'{P}\:'' \\ $$$$ \\ $$$$\varphi=\pi−\mathrm{2}\beta \\ $$$$\phi=\pi−\mathrm{2}\gamma \\ $$$$\theta=\pi−\varphi−\phi=\mathrm{2}\left(\beta+\gamma\right)−\pi=\pi−\mathrm{2}\alpha \\ $$$$\frac{{DB}}{\mathrm{sin}\:\phi}=\frac{{DC}}{\mathrm{sin}\:\varphi}=\frac{{BC}}{\mathrm{sin}\:\theta} \\ $$$$\Rightarrow{DB}=\frac{\mathrm{sin}\:\phi}{\mathrm{sin}\:\theta}×{BC}=\frac{\mathrm{sin}\:\mathrm{2}\gamma}{\mathrm{sin}\:\mathrm{2}\alpha}×{a} \\ $$$$\Rightarrow{DC}=\frac{\mathrm{sin}\:\varphi}{\mathrm{sin}\:\theta}×{BC}=\frac{\mathrm{sin}\:\mathrm{2}\beta}{\mathrm{sin}\:\mathrm{2}\alpha}×{a} \\ $$$${DP}\:'={DB}+{BP}\:'=\frac{\mathrm{sin}\:\mathrm{2}\gamma}{\mathrm{sin}\:\mathrm{2}\alpha}×{a}+\frac{{a}}{\mathrm{2}}=\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\gamma}{\mathrm{sin}\:\mathrm{2}\alpha}\right){a} \\ $$$${DP}\:''={DC}+{CP}\:''=\frac{\mathrm{sin}\:\mathrm{2}\beta}{\mathrm{sin}\:\mathrm{2}\alpha}×{a}+\frac{{a}}{\mathrm{2}}=\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\beta}{\mathrm{sin}\:\mathrm{2}\alpha}\right){a} \\ $$$$ \\ $$$${let}\:{l}={P}\:'{P}\:'' \\ $$$${l}^{\mathrm{2}} =\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\beta}{\mathrm{sin}\:\mathrm{2}\alpha}\right)^{\mathrm{2}} {a}^{\mathrm{2}} +\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\gamma}{\mathrm{sin}\:\mathrm{2}\alpha}\right)^{\mathrm{2}} {a}^{\mathrm{2}} −\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\beta}{\mathrm{sin}\:\mathrm{2}\alpha}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\gamma}{\mathrm{sin}\:\mathrm{2}\alpha}\right){a}^{\mathrm{2}} \:\mathrm{cos}\:\theta \\ $$$${l}^{\mathrm{2}} ={a}^{\mathrm{2}} \left\{\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\beta}{\mathrm{sin}\:\mathrm{2}\alpha}\right)^{\mathrm{2}} +\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\gamma}{\mathrm{sin}\:\mathrm{2}\alpha}\right)^{\mathrm{2}} +\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\beta}{\mathrm{sin}\:\mathrm{2}\alpha}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\gamma}{\mathrm{sin}\:\mathrm{2}\alpha}\right)\:\mathrm{cos}\:\mathrm{2}\alpha\right\} \\ $$$$ \\ $$$${minimum}\:{perimeter}\:{of}\:\Delta{PQR}={P}\:'{P}\:''={l} \\ $$$$={a}\sqrt{\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\beta}{\mathrm{sin}\:\mathrm{2}\alpha}\right)^{\mathrm{2}} +\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\gamma}{\mathrm{sin}\:\mathrm{2}\alpha}\right)^{\mathrm{2}} +\mathrm{2}\:\mathrm{cos}\:\mathrm{2}\alpha\:\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\beta}{\mathrm{sin}\:\mathrm{2}\alpha}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{sin}\:\mathrm{2}\gamma}{\mathrm{sin}\:\mathrm{2}\alpha}\right)} \\ $$
Commented by ajfour last updated on 09/Apr/20
$${Superb}!\:{Sir}.\:{Its}\:{smooth}\:{and}\: \\ $$$${very}\:{nice}\:{solution},\:{i}'{d}\:{delayed} \\ $$$${following}\:{it}\:{thinking}'\:{it}\:{wd}\:{be} \\ $$$${tough}\:{for}\:{me}\:{to}\:{comprehend}; \\ $$$${i}\:{was}\:{wrong},\:{thank}\:{you}\:{Sir}. \\ $$