Find-the-side-of-a-triangle-if-the-distances-from-an-arbitrary-point-inside-a-regular-triangle-to-its-vertices-are-m-n-and-k-Help-please-

Question Number 208872 by Ismoiljon_008 last updated on 26/Jun/24

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

f r o m a n a r b i t r a r y p o i n t i n s i d e a r e g u l a r t r i a n g l e

t o i t s v e r t i c e s a r e m, n a n d k .

H e l p p l e a s e

Answered by mr W last updated on 26/Jun/24

Commented by mr W last updated on 26/Jun/24

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}}

+ : p o i n t i n s i d e t h e t r i a n g l e

- : p o i n t o u t s i d e t h e t r i a n g l e

Commented by Tawa11 last updated on 26/Jun/24

Nice solution sir .

Commented by Ismoiljon_008 last updated on 27/Jun/24

t h a n k s

Answered by Ismoiljon_008 last updated on 27/Jun/24

Commented by mr W last updated on 28/Jun/24

a s h o u l d b e a s y m m e t r i c f u n c t i o n

o f v a r i a b l e s k, m, n .

Commented by Ismoiljon_008 last updated on 27/Jun/24

i s t h a t w r o n g ?