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Question Number 3943 by Rasheed Soomro last updated on 25/Dec/15

$W h a t i s t h e a r e a o f o v e r l a p p i n g r e g i o n$ $o f t h r e e c i r c l e s o f r a d i i r_{1}, r_{2}, r_{3} w i t h t h e i r$ $r e s p e c t i v e c e n t r e s C_{1}, C_{2} a n d C_{3} w h e n$ $r_{1} + r_{2} > C_{1} C_{2}, r_{2} + r_{3} > C_{2} C_{3} a n d$ $r_{3} + r_{1} > C_{3} C_{1} .$ $N o t e t h a t C_{i} C_{j} i s t h e d i s t a n c e b e t w e e n c e n t r e s$ $C_{i} a n d C_{j} .$
Commented by Yozzii last updated on 26/Dec/15

Commented by Yozzii last updated on 26/Dec/15

$Y e s . I a g r e e . L o o k a t m y i m a g e$ $f o r a b e t t e r i d e a o n f i n d i n g t h e f u l l$ $a r e a .$
Commented by Rasheed Soomro last updated on 26/Dec/15

$B u t t h e p r o b l e m i s t o d e t e r m i n e t h e c e n t r a l a n g l e s$ $o f t h e s e a r c s!$
Commented by Yozzii last updated on 25/Dec/15

$I t i s p o s s i b l e t h a t o n e c i r c l e c o m p l e t e l y$ $c o n t a i n s t h e o t h e r t w o o v e r l a p p i n g$ $c i r c l e s . T h e p r o b l e m t h e n r e q u i r e s$ $f i n d i n g o n l y t h e a r e a i n w h i c h t h e$ $t w o i n n e r c i r c l e s o v e r l a p s i n c e t h i s$ $s a m e a r e a l i e s w i t h i n t h e l a r g e r$ $c i r c l e . T h e c o n d i t i o n s o n r_{i} a n d C_{i} C_{j}$ $a r e s t i l l s a t i s f i e d .$
Commented by Yozzii last updated on 25/Dec/15

$I t h i n k I h a v e a m e a n s o f f i n d i n g t h e r e q u i r e d$ $a r e a i f n o t w o c i r c l e s f u l l y c o n t a i n e a c h$ $o t h e r . I t r e q u i r e s f o r m i n g t h r e e n e w$ $c i r c l e s a n d a t r i a n g l e . A d i a g r a m$ $w o u l d t r u l y h e l p m y e x p l a n a t i o n .$ $S u p p o s e n o t w o c i r c l e s f u l l y c o n t a i n$ $e a c h o t h e r a n d t h a t t h e y a p p e a r t o f o r m$ $e i g h t d i s t i n c t r e g i o n s i n t h e p l a n e,$ $w i t h t h e c o m m o n r e g i o n o f o v e r l a p h a v i n g t h r e e$ $v e r t i c e s A, B, C . C o n s t r u c t t h e t r i a n g l e$ $A, B, C . C a l l t h e a r e a o f △ A B C A_{1} .$ $W e h a v e A B, B C a n d C A a s a r c s,$ $b e c a u s e o f t h e m a n n e r i n w h i c h t h e$ $t h r e e c i r v l e s o v e r l a p, a n d$ $w e c a n f o r m c i r c l e s t o a d h e r e t o$ $e a c h o f t h e s e a r c s . T h u s, i f t h i s i s d o n e$ $w e c a n f i n d t h e r e s p e c t i v e a r e a s A_{2},$ $A_{3} a n d A_{4}, t h a t l i e b e t w e e n e a c h s i d e$ $o f t h e t r i a n g l e a n d t h e a r c o p p o s i t e$ $e a c h s i d e . F o r a g i v e n s i d e w i t h a n a r c,$ $t h e a r e a d i f f e r e n c e b e t w e e n t h e s e c t o r$ $o f t h e c i r c l e w i t h a r c A B, f o r e . g,$ $a n d t h e i s o s c e l e s t r i a n g l e {A B O}_{1},$ $w h e r e O_{1} i s t h e c e n t r e o f t h i s n e w$ $c i r c l e, i s A_{2}, f o r e x a m p l e . T h e a r e a o f$ $t h e r e g i o n o f o v e r l a p i s t h e n$ $A_{1} + A_{2} + A_{3} + A_{4} .$
Commented by Rasheed Soomro last updated on 25/Dec/15

$A l l p o s s i b l e c a s e s s h o u l d b e c o n s i d e r e d$ $s e p a r a t e l y$
Commented by Yozzii last updated on 25/Dec/15

$I j u s t r e a l i z e d t h a t y o u o n l y n e e d t h e$ $t h r e e i n i t i a l c i r c l e s t o d o t h e$ $c a l c u l a t i o n . T h e n e w c i r c l e s I s p e a k o f$ $o n l y n e e d t o b e t h e i n i t i a l c i r c l e s w e$ $s t a r t e d w i t h! (W a s t e o f e n e r g y . . .)$
Commented by Rasheed Soomro last updated on 25/Dec/15

$T h r e e c o p l a n e r c i r c l e s c a n f o r m a t m o s t$ $s e v e n d i s t i n c t r e g i o n s . I t h i n k .$
Commented by Yozzii last updated on 25/Dec/15

$T h e r e g i o n o u t s i d e a l l o f t h e m m a k e i t 8 .$
Commented by Rasheed Soomro last updated on 26/Dec/15

$Y o u s h o u l d s t a r t w i t h 7 - r e g i o n$ $c a s e, I t h i n k .$ $I a l s o t h i n k t h a t t h e a r e a o f o v e r l a p p i n g r e g i o n i s c l o s e l y / d i r e c l y$ $r e l a t e d t o t h e q u a n t i t i e s :$ $r_{1} + r_{2} - C_{1} C_{2}, r_{2} + r_{3} - C_{2} C_{3}, r_{3} + r_{1} - C_{3} C_{1}$ $M o r e t h e c i r c l e s a r e c l o s e d g r e a t e r t h e a r e a$ $o f o v e r l a p p i n g r e g i o n .$
Commented by Yozzii last updated on 26/Dec/15

$A, C & E a r e c i r c l e c e n t r e s . T o f i n d$ $t h e a r e a b e t w e e n t h e a r c H I a n d t h e$ $c h o r d H I, f i n d t h e a r e a o f △ C H I$ $a n d t h e n t h e a r e a o f t h e s e c t o r$ $C H I . T h e p o s i t i v e d i f f e r e n c e b e t w e e n$ $t h e t w o a r e a s g i v e s w h a t {w e}^{'} r e$ $s e a r c h i n g f o r . T h i s i s s i m i l a r l y$ $d o n e f o r s e c t o r E G H a n d s e c t o r A G I .$
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