Question-118811

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 :

{\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