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Question Number 199911 by mr W last updated on 11/Nov/23

Commented by mr W last updated on 11/Nov/23

$t h r e e m i r r o r s b u i l d t h e s i d e s o f a n$ $e q u i l a t e r a l t r i a n g l e . a l a s e r r a y$ $e m i t t e d f r o m t h e m i d p o i n t o f a s i d e$ $s h o u l d r e a c h t h e o p p o s i t e v e r t e x a f t e r$ $m u l t i p l e r e f l e c t i o n s a s a n e x a m p l e$ $i n t h e i m a g e a b o v e s h o w s .$ $f i n d t h e a n g l e α (0 < α < 90 °) .$
Commented by Frix last updated on 11/Nov/23

$I think for α < \frac{π}{2}$ $\sin α = \frac{2 k}{\sqrt{4 k^{2} + 3 {(2 n - 1)}^{2}}}$ $\sin α = \frac{2 k + 1}{\sqrt{4 k (k + 1) + 12 n^{2} + 1}}$ $with n, k \in N^{★}$
Commented by mr W last updated on 11/Nov/23

$p l e a s e r e a s o n y o u r a n s w e r a n d g i v e$ $a c o n c r e t e e x a m p l e a n d v e r i f y i t,$ $f o r e x a m p l e t h e c a s e a s s h o w n i n$ $t h e i m a g e . t h a n k s!$
Commented by Frix last updated on 11/Nov/23

$We can draw a tetragonal grid and mark$ $all reflections of the target vertice . All$ $possible paths are straight lines within the$ $grid . The horizontal of the reflections with$ $the start point set to (0, 0) are$ $(\frac{3}{2} \times (2 n - 1), \sqrt{3} k) or (\frac{3}{2} \times 2 n, \frac{\sqrt{3}}{2} + \sqrt{3} k)$ $I might be wrong, writing this while on my$ $way to work and without any possibility$ $to draw it .$
Commented by Frix last updated on 11/Nov/23

$I think the image shows the first case with$ $k = 1 \land n = 2$ $\sin α = \frac{2}{\sqrt{31}} \Rightarrow \tan α = \frac{2 \sqrt{3}}{9} \Rightarrow α \approx 21.05 °$
Commented by mr W last updated on 11/Nov/23

$c o r r e c t!$
	Answered by ajfour last updated on 11/Nov/23

	Answered by ajfour last updated on 11/Nov/23

	$α = \tan^{- 1} (\frac{2}{\sqrt{3}})$
	Commented by mr W last updated on 12/Nov/23

	$y e s, t h a n k s s i r!$
	Answered by mr W last updated on 11/Nov/23

	Commented by mr W last updated on 11/Nov/23

	$r e d m i r r o r i s t h e f i r s t m i r r o r w h i c h$ $r e f l e c t s t h e l i g h t r a y a n d t h e b r o w n$ $o n e i s t h e l a s t m i r r o r . b e t w e e n t h e m$ $a l l t h r e e m i r r o r s m a y b e h i t b y t h e l i g h t$ $r a y o n e o r m o r e t i m e s .$ $w e c a n f i n d t h a t i n t h e r e d m i r r o r$ $t h e d e s t i n a t i o n p o i n t (r e d d o t) m u s t$ $l i e o n t h e b l u e l i n e a n d t h e p o s s i b l e$ $p o s i t i o n s a r e b l u e m a r k e d .$ $w e c a n f i n d o u t t h a t t h e a n g l e α i s$ $α = \tan^{- 1} \frac{2 \sqrt{3}}{n} w h e r e n i s a n o d d n u m b e r$ $w h i c h i s e q u a l t o o r l a r g e r t h a n 3 .$ $n m e a n s a l s o h o w m a n y t i m e s t h e$ $l i g h t r a y i s r e f l e c t e d b e f o r e i t r e a c h e s$ $t h e d e s t i n a t i o n .$
	Commented by mr W last updated on 11/Nov/23

	Commented by mr W last updated on 11/Nov/23

	$t h i s e x a m p l e s h o w s w h e n t h e l i g h t$ $i s r e f l e c t e d 3 t i m e s .$
	Commented by mr W last updated on 11/Nov/23

	Commented by mr W last updated on 11/Nov/23

	$t h i s e x a m p l e s h o w s t h e c a s e g i v e n$ $i n t h e q u e s t i o n w h e r e t h e l i g h t r a y$ $i s r e f l e c t e d 9 t i m e s .$
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