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Question Number 52261 by mr W last updated on 05/Jan/19

Commented by mr W last updated on 05/Jan/19

$T h r e e c i r c l e s w i t h r a d i i r_{a}, r_{b} a n d r_{c}$ $h a v e t h e i r c e n t e r s a t A, B a n d C w h i c h$ $h a v e d i s t a n c e s a, b a n d c t o e a c h o t h e r .$ $F i n d t h e m a x i m u m a n d m i n i m u m$ $a r e a o f t h e t r i a n g l e w h o s e v e r t e x e s$ $a r e o n e a c h o f t h e c i r c l e s .$
Commented by behi83417@gmail.com last updated on 05/Jan/19

$i t h i n k : m a x o r m i n o f a r e a h a p p e n s$ $w h e n t h e s i d e s o f P \overset{△}{Q R} a r e p a r a l l e l t o$ $s i d e s o f A \overset{△}{B C} .$
Commented by mr W last updated on 06/Jan/19

$s i n c e t h e c i r c l e s h a v e d i f f e r e n t r a d i i,$ $i t i s n o t a l w a y s p o s s i b l e t o b u i l d a t r i a n g l e$ $P Q R w h o s e s i d e s a r e p a r a l l e l t o t h e$ $s i d e s f r o m t r i a n g l e A B C .$
Commented by ajfour last updated on 05/Jan/19

Commented by ajfour last updated on 05/Jan/19

$s h a l l a t t e m p t l a t e r, S i r .$
Commented by mr W last updated on 06/Jan/19

$t h a n k y o u s i r . {i t}^{'} s c o r r e c t :$ $t h e m a x i m u m (a s w e l l a s t h e m i n i m u m)$ $t r i a n g l e P Q R s h o u l d b e c o n s t r u c t e d$ $i n t h e w a y t h a t t h e t a n g e n t a t R i s$ $p a r a l l e l t o t h e s i d e P Q a n d t h e t a n g e n t$ $a t P i s p a r a l l e l t o t h e s i d e Q R a n d t h e$ $t a n g e n t a t Q i s p a r a l l e l t o t h e s i d e R P .$ $i . e . C R ⊥ P Q, A P ⊥ Q R, B Q ⊥ R P .$
Commented by ajfour last updated on 06/Jan/19

$A (0, 0), r_{A} = α, r_{B} = β, r_{C} = γ$ $C (\frac{b^{2} + c^{2} - a^{2}}{2 c}, \sqrt{b^{2} - {(\frac{b^{2} + c^{2} - a^{2}}{2 c})}^{2}})$ $P (- α \cos ϕ, - α \sin ϕ)$ $Q (c + β \cos θ, - β \sin θ)$ $m_{P Q} = \tan δ = \frac{β \sin θ - α \sin ϕ}{c + β \cos θ + α \cos ϕ}$ $R (x_{C} + γ \sin δ, y_{C} + γ \cos δ)$ $m_{P R} = \frac{1}{\tan θ}, m_{Q R} = - \frac{1}{\tan ϕ}$ $e q . o f P R : y - y_{P} = m_{P R} (x - x_{P})$ $e q . o f Q R : y - y_{Q} = m_{Q R} (x - x_{Q})$ $T h e i r i n t e r s e c t i o n g i v e s R (x_{R}, y_{R})$ $x_{R} = x_{C} + γ \sin δ$ $y_{R} = y_{C} + γ \cos δ$ $H e n c e θ, ϕ a r e d e t e r m i n e d .$ $r^{2} = {P Q}^{2} = {(β \sin θ - α \sin ϕ)}^{2}$ $+ {(c + α \cos θ + β \cos ϕ)}^{2}$ $A_{m a x}^{2} = \frac{r^{2} h^{2}}{4}$ $e q . o f P Q : y - y_{P} = \tan δ (x - x_{P})$ $h = γ + \frac{y_{C} - y_{P} - \tan δ (x_{C} - x_{P})}{\sqrt{1 + \tan^{2} δ}} .$ $F o r A_{m i n}, w e g o a s i m i l a r w a y w i t h$ $α \to - α, β \to - β, γ \to - γ .$
Commented by ajfour last updated on 06/Jan/19

$Y e s S i r, t h a n k y o u . S p l e n d i d q u e s t i o n!$
Commented by mr W last updated on 06/Jan/19

$p r o c e s s i s c o r r e c t s i r!$ $v e r y n i c e .$
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