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Question Number 121615 by ajfour last updated on 10/Nov/20

Commented by ajfour last updated on 10/Nov/20

$$Thebluetriangleisrightangledand$$$$isosceleswithAF=BF=3.$$$$Radiusofcircler=2.$$$$Findmaximumsidelengthof$$$$equilateral△DEF.$$

Commented by MJS_new last updated on 10/Nov/20

$$\left(1\right)\mathrm{start}\mathrm{with}\mathrm{the}\mathrm{blue}\mathrm{triangle}$$$$\left(2\right)D\mathrm{on}\mathrm{line}AB$$$$\left(3\right)\mathrm{construct}E$$$$\Rightarrow \mathrm{we}\mathrm{have}\mathrm{the}\mathrm{minimal}\mathrm{side}s\mathrm{when}D=\frac{A+B}{2}$$$$\Rightarrow \mathrm{the}\mathrm{maximum}\mathrm{depends}\mathrm{on}\mathrm{the}\mathrm{possibility}$$$$\mathrm{of}\mathrm{the}\mathrm{circle}\mathrm{which}\mathrm{exists}\mathrm{for}\mid AE\mid ⩽4$$$$\mathrm{if}D\mathrm{is}\mathrm{between}A\mathrm{and}B\mathrm{and}\mathrm{closer}\mathrm{to}A$$$$\mathrm{the}\mathrm{maximum}\mathrm{is}\mathrm{at}D=A$$$$\mathrm{if}\mathrm{we}\mathrm{allow}D\mathrm{anywhere}\mathrm{on}\mathrm{the}\mathrm{line}\mathrm{through}$$$$A\mathrm{and}B{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{harder}\mathrm{to}\mathrm{calculate}...$$

Commented by ajfour last updated on 10/Nov/20

$$wontbeastoughsir,pleasearrive$$$$atananswer,ishalltrythenormal$$$$easyway,howsoeverlengthy..$$

Commented by MJS_new last updated on 10/Nov/20

$$\mathrm{ok}\mathrm{but}\mathrm{later}$$

Commented by ajfour last updated on 10/Nov/20

$$ithinkfor{s}_{max}DisatA,s_{max}=3.$$

Commented by MJS_new last updated on 10/Nov/20

$$\mathrm{yes}.\mathrm{if}D\mathrm{is}\mathrm{allowed}\mathrm{only}\mathrm{between}A\mathrm{and}B.$$$$\mathrm{else}{s}_{max}=4$$

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