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Question Number 118811 by ajfour last updated on 19/Oct/20

Commented by ajfour last updated on 19/Oct/20

$W h a t i s t h e s i d e l e n g t h o f t h e e q u a l$ $s i d e d h e x a g o n i n s c r i b e d w i t h i n t h e$ $t r i a n g l e A B C a s s h o w n ?$
Commented by I want to learn more last updated on 20/Oct/20

$MrW, please come and solve .$
Commented by mr W last updated on 21/Oct/20

$a j f o u r s i r :$ $i t h i n k t h e r e i s n o u n i q u e s o l u t i o n .$
Commented by ajfour last updated on 21/Oct/20

$f i n e s i r, h o w a b o u t e x t r e m e v a l u e s ?$
	Answered by 1549442205PVT last updated on 20/Oct/20

	$This problem has infinite many of$ $solutions in case the triangle ABC$ $is equilateral since we can construct$ $an infinite number of hexagons with$ $equal sides and side length is different$
	Commented by ajfour last updated on 20/Oct/20

	$n o, y o u d o n t f o l l o w; g i v e n t h a t s i d e s$ $o f t r i a n g l e a r e A B = c, B C = a, C A = b;$ $f i n d s i d e s o f t h e h e x a g o n i n t e r m s$ $o f a, b, c .$
	Commented by 1549442205PVT last updated on 20/Oct/20

	Commented by 1549442205PVT last updated on 20/Oct/20
	$Put AD=m,AE=n,CM=p,CH=q,ME=t BF=r,BG=s,BC=a,AC=b,AB=c From the cosine theorem we have : { ((t^2 =m^2 +n^2 −2mncosA(1))),((t^2 =p^2 +q^2 −2pqcosC(2))),((t^2 =r^2 +s^2 −2rscosB(3))),((n+t+p=b(4))),((q+t+r=a(5))),((m+t+s=c(6))) :} Thus,we have the system of 6 equations with 6 unknowns where a,b,c,cosA, cosB,cosC are parameters which are independent together and now we need have to solve it.$
	$Put AD = m, AE = n, CM = p, CH = q, ME = t$ $BF = r, BG = s, BC = a, AC = b, AB = c$ $From the cosine theorem we have :$ ${\begin{cases} t^{2} = m^{2} + n^{2} - 2 mncosA (1) \\ t^{2} = p^{2} + q^{2} - 2 pqcosC (2) \\ t^{2} = r^{2} + s^{2} - 2 rscosB (3) \\ n + t + p = b (4) \\ q + t + r = a (5) \\ m + t + s = c (6) \end{cases}$ $Thus, we have the system of 6 equations$ $with 6 unknowns where a, b, c, cosA,$ $cosB, cosC are parameters which$ $are independent together and now$ $we need have to solve it .$
	Commented by I want to learn more last updated on 20/Oct/20

	$Sir, 1549442205 PVT which app you use to draw .$
	Commented by 1549442205PVT last updated on 21/Oct/20

	$Geogebra, Drawing of tinkutara done this$
	Commented by mr W last updated on 21/Oct/20

	$6 e q u a t i o n s, b u t 7 u n k n o w n s,$ $d o e s i t m e a n t h a t t h e r e i s n o u n i q u e$ $s o l u t i o n ?$
	Commented by 1549442205PVT last updated on 25/Oct/20

	$Yes, Sir is right . I also have such thinkings$
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