Question-123040

Question Number 123040 by ajfour last updated on 22/Nov/20

Commented by ajfour last updated on 22/Nov/20

F i n d m a x i m u m s i d e l e n g t h o f

e q u i l a t e r a l △ A B C, i n t e r m s o f

p, q, a n d r; t h e v e r t i c e s o f

w h i c h l i e r e s p e c t i v e l y o n t h r e e

c o n c e n t r i c c i r c l e s o f r a d i i

p < q < r . A s s u m e v e r t e x A i s a s

s h o w n o n o u t e r m o s t c i r c l e a n d o n

v e r t i c a l a x e s .

Commented by mr W last updated on 22/Nov/20

w e g e t t w o t r i a n g l e s w i t h s i d e l e n g t h

s = \sqrt{\frac{p^{2} + q^{2} + r^{2} \pm \sqrt{3 δ}}{2}} w i t h

δ = (p + q + r) (- p + q + r) (p - q + r) (p + q - r)

Commented by ajfour last updated on 23/Nov/20

Commented by mr W last updated on 23/Nov/20

g r e a t!

Answered by mr W last updated on 22/Nov/20

Commented by ajfour last updated on 23/Nov/20

T h a n k s a l o t s i r .

Commented by mr W last updated on 22/Nov/20

B (- \frac{s}{2}, 0)

C (\frac{s}{2}, 0)

A (0, \frac{\sqrt{3} s}{2})

G (h, k) s a y

h^{2} + {(k - \frac{\sqrt{3} s}{2})}^{2} = p^{2} \dots (i)

{(h + \frac{s}{2})}^{2} + k^{2} = q^{2} \dots (i i)

{(h - \frac{s}{2})}^{2} + k^{2} = r^{2} \dots (i i i)

(i i) - (i i i) :

\Rightarrow 2 h s = q^{2} - r^{2}

(i i) - (i) :

h s + \frac{s^{2}}{4} + \sqrt{3} k s - \frac{3 s^{2}}{4} = q^{2} - p^{2}

\Rightarrow 2 \sqrt{3} k s = s^{2} + q^{2} + r^{2} - 2 p^{2}

i n t o (i i i) :

3 {(q^{2} - r^{2} - s^{2})}^{2} + {(s^{2} + q^{2} + r^{2} - 2 p^{2})}^{2} = 12 r^{2} s^{2}

s^{4} - (p^{2} + q^{2} + r^{2}) s^{2} + p^{4} + q^{4} + r^{4} - p^{2} q^{2} - q^{2} r^{2} - r^{2} p^{2} = 0

s^{2} = \frac{p^{2} + q^{2} + r^{2} \pm \sqrt{{(p^{2} + q^{2} + r^{2})}^{2} - 4 (p^{4} + q^{4} + r^{4} - p^{2} q^{2} - q^{2} r^{2} - r^{2} p^{2})}}{2}

= \frac{p^{2} + q^{2} + r^{2} \pm \sqrt{6 (p^{2} q^{2} + q^{2} r^{2} + r^{2} p^{2}) - 3 (p^{4} + q^{4} + r^{4})}}{2}

= \frac{p^{2} + q^{2} + r^{2} \pm \sqrt{3 (p + q + r) (- p + q + r) (p - q + r) (p + q - r)}}{2}

\Rightarrow s = \sqrt{\frac{p^{2} + q^{2} + r^{2} \pm \sqrt{3 (p + q + r) (- p + q + r) (p - q + r) (p + q - r)}}{2}}

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