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

$l e t A B C a g i v e n t r i a n g l e . C a n w e f i n d$ $t h r e e p o s i t i o n s I, J, K o n t h e s i d e A B, A C, B C$ $S u c h t h a t I J K i s e q u i l a t e r a l ?$
Commented by mr W last updated on 01/Feb/24

$y e s, w e c a n f i n d i n f i n i t e m a n y s u c h$ $p o i n t s . t h e s m a l l e s t e q u i l a t e r a l$ $t r i a n g l e I J K h a s t h e s i d e l e n g t h$ $s = \frac{2 \sqrt{2} Δ}{\sqrt{a^{2} + b^{2} + c^{2} + 4 \sqrt{3} Δ}} w i t h$ $Δ = a r e a o f A B C .$
Commented by mr W last updated on 01/Feb/24

$f o l l o w i n g i m a g e s h o w s h o w y o u$ $c a n f i n d s u c h a n e q u i l a t e r a l t r i a n g l e :$
Commented by mr W last updated on 01/Feb/24

Commented by Rasheed.Sindhi last updated on 02/Feb/24

$Sir, h o w c a n y o u s a y t h e r e a r e$ $i n f i n i t e s u c h p o i n t s ? T h e a b o v e$ $p r o c e s s c a n b e a p p l i e d e x a c t l y$ $t h r e e w a y s, I t h i n k .$
Commented by mr W last updated on 03/Feb/24

$t h e p r o c e s s a b o v e s h o w s o n l y o n e w a y$ $h o w t o g e t t h r e e o f s u c h i n s c r i b e d$ $e q u i l a t e r a l t r i a n g l e s . b u t t h e r e a r e$ $m a n y m a n y o t h e r s u c h e q u i l a t e r a l$ $t r i a n g l e s .$ $s a y t h e i n t e r i o r a n g l e s o f t h e g i v e n$ $t r i a n g l e A B C a r e α, β, γ . f o l l o w i n g$ $i m a g e s h o w s t h a t a g i v e n e q u i l a t e r a l$ $t r i a n g l e m a y h a v e i n f i n i t e l y m a n y$ $c i r c u m t r i a n g l e s w h o s e i n t e r i o r$ $a n g l e s a r e α, β, γ . a l l t h e s e c i r c u m -$ $t r i a n g l e s a r e s i m i l a r t o t h e o r i g i n a l$ $t r i a n g l e A B C . t h a t m e a n s r e v e r s e l y$ $t h a t t h e t r i a n g l e A B C m a y h a v e$ $i n f i n i t e l y m a n y i n s c r i b e d e q u i l a t e r a l$ $t r i a n g l e s .$
Commented by mr W last updated on 03/Feb/24

Commented by mr W last updated on 02/Feb/24

$s e e Q 82131 f o r m o r e a b o u t t h i s t o p i c .$
Commented by Rasheed.Sindhi last updated on 03/Feb/24

$T h a n k s a l o t s i r! I u n d e r s t o o d$ $c l e a r l y .$
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