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Question Number 16194 by RasheedSoomro last updated on 18/Jun/17

Commented by RasheedSoomro last updated on 18/Jun/17

$$\mathrm{A},\mathrm{B}\mathrm{and}\mathrm{C}\mathrm{are}\mathrm{three}\mathrm{non}-\mathrm{collinear}\mathrm{points}.$$$$\mathrm{Also}\mathrm{they}\mathrm{are}\mathrm{centers}\mathrm{of}\mathrm{three}\mathrm{circles}\mathrm{having}$$$$\mathrm{radii}a,b\mathrm{and}c\mathrm{respectively}.\mathrm{D},\mathrm{E}\mathrm{\&}\mathrm{F}\mathrm{have}\mathrm{been}$$$$\mathrm{taken}\mathrm{on}\mathrm{three}\mathrm{circles}\mathrm{respectively}\mathrm{in}\mathrm{such}\mathrm{a}\mathrm{way}$$$$\mathrm{that}△\mathrm{DEF}\mathrm{is}\mathrm{an}\mathrm{equilateral}\mathrm{triangle}.$$$$\left(\mathrm{i}\right)\mathrm{Draw}\mathrm{any}\mathrm{such}\mathrm{triangle}\mathrm{using}\mathrm{Euclidean}\mathrm{tools}.$$$$\left(\mathrm{ii}\right)\left(\mathrm{Using}\mathrm{Euclidean}\mathrm{tools}\right)\mathrm{Draw}\mathrm{such}\mathrm{triangle}\mathrm{of}$$$$\mathrm{maximum}\mathrm{size}\mathrm{and}\mathrm{find}\mathrm{out}\mathrm{its}\mathrm{area}.$$$$\left(\mathrm{ii}\right)\left(\mathrm{Using}\mathrm{Euclidean}\mathrm{tools}\right)\mathrm{Draw}\mathrm{such}\mathrm{triangle}\mathrm{of}$$$$\mathrm{minimum}\mathrm{size}\mathrm{and}\mathrm{find}\mathrm{out}\mathrm{its}\mathrm{area}.$$$$\mathrm{Take}\mathrm{AB}=l,\mathrm{BC}=m\mathrm{and}\mathrm{CA}=n.$$

Commented by RasheedSoomro last updated on 18/Jun/17

$$\mathrm{I}\mathrm{think}\mathrm{the}\mathrm{question}\mathrm{is}\mathrm{clear}\mathrm{now}.$$$$\mathrm{In}\mathrm{question}\mathrm{of}\mathrm{Mr}\mathrm{Amir}\left(\mathrm{Behi}\right)\mathrm{the}\mathrm{circles}$$$$\mathrm{were}\mathrm{concentric},\mathrm{whereas}\mathrm{here}\mathrm{circles}\mathrm{have}$$$$\mathrm{different}\mathrm{centers}.$$

Commented by mrW1 last updated on 20/Jun/17

$$\mathrm{This}\mathrm{is}\mathrm{a}\mathrm{very}\mathrm{interesting}\mathrm{and}\mathrm{challenging}$$$$\mathrm{question},\mathrm{sir}.\mathrm{Its}\mathrm{solution}\mathrm{would}\mathrm{be}$$$$\mathrm{not}\mathrm{so}\mathrm{easy}\mathrm{I}\mathrm{think}.$$$$\mathrm{For}\mathrm{a}\mathrm{fixed}\mathrm{point}\mathrm{on}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{three}$$$$\mathrm{circles}\mathrm{we}\mathrm{can}\mathrm{find}\mathrm{the}\mathrm{equilateral}$$$$\mathrm{triangle}\left(\mathrm{s}\right)\mathrm{in}\mathrm{the}\mathrm{same}\mathrm{way}\mathrm{as}\mathrm{I}$$$$\mathrm{described}\mathrm{in}\mathrm{Q}15969.\mathrm{Then}\mathrm{we}\mathrm{change}$$$$\mathrm{the}\mathrm{position}\mathrm{of}\mathrm{the}\mathrm{fixed}\mathrm{point}\mathrm{along}$$$$\mathrm{the}\mathrm{first}\mathrm{circle},\mathrm{the}\mathrm{size}\mathrm{of}\mathrm{the}\mathrm{equilateral}$$$$\mathrm{triangle}\left(\mathrm{s}\right)\mathrm{will}\mathrm{change}\mathrm{correspondingly}.$$$$\mathrm{Among}\mathrm{them}\mathrm{to}\mathrm{find}\mathrm{the}\mathrm{minimal}\mathrm{and}$$$$\mathrm{maximal}\mathrm{one}\mathrm{will}\mathrm{be}\mathrm{very}\mathrm{hard}.$$

Commented by RasheedSoomro last updated on 19/Jun/17

$$\mathcal{THANKS}\mathcal{SIR}!$$$$\mathrm{Is}\mathrm{there}\mathrm{any}\mathrm{analytical}\mathrm{approach}\mathrm{for}\mathrm{finding}$$$$\mathrm{maximum}/\mathrm{minimum}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{triangle}?$$

Commented by mrW1 last updated on 19/Jun/17

Commented by mrW1 last updated on 20/Jun/17

$$\mathrm{This}\mathrm{should}\mathrm{be}\mathrm{an}\mathrm{animated}\mathrm{GIF}\mathrm{image}.$$$$\mathrm{Unfortunately}\mathrm{it}\mathrm{is}\mathrm{not}\mathrm{supported}\mathrm{by}$$$$\mathrm{the}\mathrm{forum}.\mathrm{It}\mathrm{shows}\mathrm{how}\mathrm{the}\mathrm{equi}-$$$$\mathrm{lateral}\mathrm{triangles}\mathrm{change}\mathrm{when}\mathrm{the}\mathrm{point}$$$$\mathrm{D}\mathrm{varies},\mathrm{see}\mathrm{also}\mathrm{single}\mathrm{pictures}\mathrm{later}$$$$\mathrm{for}\mathrm{different}\mathrm{positions}\mathrm{of}\mathrm{point}\mathrm{D}.$$$$$$$$\mathrm{The}\mathrm{position}\mathrm{of}\mathrm{point}\mathrm{D}\mathrm{can}\mathrm{be}$$$$\mathrm{expressed}\mathrm{as}\mathrm{a}\mathrm{variable},\mathrm{for}\mathrm{example}$$$$\theta .\mathrm{The}\mathrm{size}\mathrm{of}\mathrm{the}\mathrm{triangle}\left(\mathrm{s}\right)\mathrm{can}\mathrm{also}$$$$\mathrm{be}\mathrm{expressed}\mathrm{as}\mathrm{a}\mathrm{function}\mathrm{of}\theta :$$$$\mathrm{Side}\mathrm{length}\mathrm{of}\mathrm{triangle}=\mathrm{s}\left({\mathrm{s}}_1\mathrm{and}{\mathrm{s}}_2\right)$$$$\mathrm{s}=\mathrm{f}\left(\theta \right)$$$$\mathrm{The}\max .\mathrm{and}\min .\mathrm{of}\mathrm{s}\mathrm{can}\mathrm{be}\mathrm{obtained}$$$$\mathrm{by}\frac{\mathrm{ds}}{\mathrm{d}\theta }=0.$$$$\mathrm{The}\mathrm{theory}\mathrm{is}\mathrm{easy},\mathrm{the}\mathrm{doing}\mathrm{is}\mathrm{not},\mathrm{but}$$$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{generally}\mathrm{possible}.$$

Commented by mrW1 last updated on 19/Jun/17

Commented by mrW1 last updated on 19/Jun/17

Commented by mrW1 last updated on 19/Jun/17

Commented by mrW1 last updated on 19/Jun/17

Commented by mrW1 last updated on 19/Jun/17

Commented by mrW1 last updated on 19/Jun/17

Commented by mrW1 last updated on 19/Jun/17

Commented by RasheedSoomro last updated on 20/Jun/17

$$\mathbf{Oh}\mathbf{Sir},\mathbf{lot}\mathbf{of}\mathcal{T}\mathcal{HANKS}!!$$

Commented by mrW1 last updated on 20/Jun/17

Commented by mrW1 last updated on 20/Jun/17

$$\mathrm{When}\mathrm{we}\mathrm{display}\mathrm{the}\mathrm{size}\mathrm{of}\mathrm{the}\mathrm{equi}-$$$$\mathrm{lateral}\mathrm{triangles}\mathrm{corresponding}\mathrm{to}$$$$\mathrm{point}\mathrm{D}\mathrm{as}{\mathrm{AS}}_1\mathrm{and}{\mathrm{AS}}_2,\mathrm{we}\mathrm{will}\mathrm{get}$$$$\mathrm{two}\mathrm{curves}\mathrm{for}\mathrm{point}{\mathrm{S}}_1\mathrm{and}{\mathrm{S}}_2,$$$$\mathrm{which}\mathrm{show}\mathrm{the}\mathrm{change}\mathrm{of}\mathrm{the}\mathrm{size}\mathrm{of}$$$$\mathrm{the}\mathrm{triangles}\mathrm{when}\mathrm{point}\mathrm{D}\mathrm{changes}.$$$$\mathrm{Then}\mathrm{we}\mathrm{can}\mathrm{get}\mathrm{the}\min .\mathrm{and}\max .$$$$\mathrm{of}\mathrm{them},\mathrm{see}\mathrm{diagram}.$$$$\mathrm{AN}=\min .\mathrm{side}\mathrm{length}\mathrm{of}\mathrm{triangle}$$$$\mathrm{AM}=\max .\mathrm{side}\mathrm{length}\mathrm{of}\mathrm{triangle}$$

Commented by RasheedSoomro last updated on 20/Jun/17

$$\mathrm{One}\mathrm{thing}\mathrm{to}\mathrm{understand}:$$$$\mathrm{S}1\mathrm{and}\mathrm{S}2\mathrm{are}\mathrm{points}\mathrm{and}\mathrm{also}\mathrm{lengths}?$$

Commented by mrW1 last updated on 20/Jun/17

$$\overline{{\mathrm{AS}}_1}=\mathrm{length}{\mathrm{s}}_1$$$$\overline{{\mathrm{AS}}_2}=\mathrm{length}{\mathrm{s}}_2$$

Commented by mrW1 last updated on 20/Jun/17

Commented by mrW1 last updated on 20/Jun/17

Commented by RasheedSoomro last updated on 21/Jun/17

$$\mathcal{TH}\mathit{\alpha }\mathit{n}\mathit{k}\mathit{s}foryourdeeplyinvolvement$$$$inmyquestion!$$$$\mathcal{TH}\mathit{\alpha }\mathit{n}\mathit{k}\mathit{s}forsomuchlabour!$$

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