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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-




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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