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let-ABC-a-given-triangle-Can-we-find-three-positions-I-J-K-on-the-side-AB-AC-BC-Such-that-IJK-is-equilateral-

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Question Number 203905 by sniper237 last updated on 01/Feb/24
let ABC a given triangle. Can we find three positions I,J,K on the side AB,AC,BC Such that IJK is equilateral?
$${let}\:{ABC}\:{a}\:{given}\:{triangle}.\:{Can}\:{we}\:{find} \\ $$$${three}\:{positions}\:{I},{J},{K}\:{on}\:{the}\:{side}\:{AB},{AC},{BC} \\ $$$${Such}\:{that}\:{IJK}\:{is}\:{equilateral}? \\ $$
Commented by mr W last updated on 01/Feb/24
yes, we can find infinite many such points. the smallest equilateral triangle IJK has the side length s=((2(√2)Δ)/( (√(a^2 +b^2 +c^2 +4(√3)Δ)))) with Δ=area of ABC.
$${yes},\:{we}\:{can}\:{find}\:{infinite}\:{many}\:{such} \\ $$$${points}.\:{the}\:{smallest}\:{equilateral} \\ $$$${triangle}\:{IJK}\:{has}\:{the}\:{side}\:{length} \\ $$$${s}=\frac{\mathrm{2}\sqrt{\mathrm{2}}\Delta}{\:\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\mathrm{4}\sqrt{\mathrm{3}}\Delta}}\:{with} \\ $$$$\Delta={area}\:{of}\:{ABC}. \\ $$
Commented by mr W last updated on 01/Feb/24
following image shows how you can find such an equilateral triangle:
$${following}\:{image}\:{shows}\:{how}\:{you} \\ $$$${can}\:{find}\:{such}\:{an}\:{equilateral}\:{triangle}: \\ $$
Commented by mr W last updated on 01/Feb/24
Commented by Rasheed.Sindhi last updated on 02/Feb/24
Sir, how can you say there are infinite such points? The above process can be applied exactly three ways, I think.
$$\boldsymbol{\mathrm{Sir}},\:{how}\:{can}\:{you}\:{say}\:{there}\:{are} \\ $$$$\boldsymbol{{infinite}}\:{such}\:{points}?\:{The}\:{above} \\ $$$${process}\:{can}\:{be}\:{applied}\:{exactly} \\ $$$${three}\:{ways},\:{I}\:{think}. \\ $$
Commented by mr W last updated on 03/Feb/24
the process above shows only one way how to get three of such inscribed equilateral triangles. but there are many many other such equilateral triangles. say the interior angles of the given triangle ABC are α, β, γ. following image shows that a given equilateral triangle may have infinitely many circumtriangles whose interior angles are α, β, γ. all these circum− triangles are similar to the original triangle ABC. that means reversely that the triangle ABC may have infinitely many inscribed equilateral triangles.
$${the}\:{process}\:{above}\:{shows}\:{only}\:{one}\:{way} \\ $$$${how}\:{to}\:{get}\:{three}\:{of}\:{such}\:{inscribed}\: \\ $$$${equilateral}\:{triangles}.\:{but}\:{there}\:{are} \\ $$$${many}\:{many}\:{other}\:{such}\:{equilateral} \\ $$$${triangles}. \\ $$$${say}\:{the}\:{interior}\:{angles}\:{of}\:{the}\:{given} \\ $$$${triangle}\:{ABC}\:{are}\:\alpha,\:\beta,\:\gamma.\:{following}\: \\ $$$${image}\:{shows}\:{that}\:{a}\:{given}\:{equilateral} \\ $$$${triangle}\:{may}\:{have}\:{infinitely}\:{many} \\ $$$${circumtriangles}\:{whose}\:{interior} \\ $$$${angles}\:{are}\:\alpha,\:\beta,\:\gamma.\:{all}\:{these}\:{circum}− \\ $$$${triangles}\:{are}\:{similar}\:{to}\:{the}\:{original} \\ $$$${triangle}\:{ABC}.\:{that}\:{means}\:{reversely}\: \\ $$$${that}\:{the}\:{triangle}\:{ABC}\:{may}\:{have} \\ $$$${infinitely}\:{many}\:{inscribed}\:{equilateral} \\ $$$${triangles}. \\ $$
Commented by mr W last updated on 03/Feb/24
Commented by mr W last updated on 02/Feb/24
see Q82131 for more about this topic.
$${see}\:{Q}\mathrm{82131}\:{for}\:{more}\:{about}\:{this}\:{topic}. \\ $$
Commented by Rasheed.Sindhi last updated on 03/Feb/24
Thanks a lot sir! I understood clearly.
$$\boldsymbol{\mathcal{T}{hanks}}\:\boldsymbol{{a}}\:\boldsymbol{{lot}}\:\boldsymbol{{sir}}!\:\boldsymbol{{I}}\:\boldsymbol{{understood}} \\ $$$$\boldsymbol{{clearly}}. \\ $$

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