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Question Number 79675 by ajfour last updated on 27/Jan/20

Commented by ajfour last updated on 27/Jan/20

$$Thesquarehassides.$$$$Canwefindasetofradiivalues?$$

Commented by ajfour last updated on 27/Jan/20

Commented by ajfour last updated on 27/Jan/20

$$Thankyousir!goodquestion,$$$$ithinktoo.$$

Commented by mr W last updated on 27/Jan/20

$$letc=\lambda s$$$${(a+c)}^2-{(a-c)}^2={(s-c)}^2$$$$4ac={(s-c)}^2$$$$\Rightarrow a=\frac{{(s-c)}^2}{4c}=\frac{{(1-\lambda )}^{2}s}{4\lambda }$$$$$$$${(b+c)}^2-{(s-b-c)}^2=c^2$$$$2bs={(s-c)}^2$$$$\Rightarrow b=\frac{{(s-c)}^2}{2s}=\frac{{(1-\lambda )}^{2}s}{2}$$$$$$$$a+b=s$$$$\Rightarrow \frac{{(1-\lambda )}^2}{4\lambda }+\frac{{(1-\lambda )}^2}{2}=1$$$$\Rightarrow {(1-\lambda )}^2(2\lambda +1)=4\lambda$$$$\Rightarrow 2{\lambda }^3-3{\lambda }^2-4\lambda +1=0$$$$\Rightarrow (\lambda +1)(2{\lambda }^2-5\lambda +1)=0$$$$\Rightarrow \lambda =-1<0$$$$\Rightarrow \lambda =\frac{5\pm \sqrt{17}}{4}should<1$$$$\Rightarrow thereisonlyonesetofradiiwith$$$$\lambda =\frac{5-\sqrt{17}}{4}\approx 0.219$$$$a=\frac{(7+\sqrt{17})s}{16}\approx 0.695s$$$$b=\frac{(9-\sqrt{17})s}{16}\approx 0.305s$$$$c=\frac{(5-\sqrt{17})s}{4}\approx 0.219s$$

Commented by mr W last updated on 27/Jan/20

$$nicequestion!$$

Commented by MJS last updated on 27/Jan/20

$$\mathrm{can}\mathrm{we}\mathrm{find}\mathrm{a}\mathrm{solution}\mathrm{when}AB\mathrm{is}\mathrm{not}\mathrm{parallel}$$$$\mathrm{to}s?\mathrm{it}\mathrm{might}\mathrm{get}\mathrm{hard}...$$

Commented by ajfour last updated on 27/Jan/20

$$itcameoutautomatically,$$$$questiondidnotgivesucha$$$$condition,Sir.$$

Commented by mr W last updated on 27/Jan/20

$$sincecentersAandBareonthe$$$$oppositesides,wehaveautomatically$$$$a+b=s.$$

Commented by MJS last updated on 27/Jan/20

$$\mathrm{ok}$$

Commented by mr W last updated on 27/Jan/20

$$MJSsir:doyouhaveexperiencewith$$$$trilinearcoordinatessystem?canit$$$$behelpfultosolveQ79694?thanks!$$

Commented by MJS last updated on 27/Jan/20

$$\mathrm{sorry}\mathrm{I}{\mathrm{can}}^{\prime }\mathrm{t}\mathrm{help}$$

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