prove that the quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.


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Question Number 10667 by Gaurav3651 last updated on 22/Feb/17

$p r o v e t h a t t h e q u a d r i l a t e r a l f o r m e d$ $b y a n g l e b i s e c t o r s o f a c y c l i c$ $q u a d r i l a t e r a l i s a l s o c y c l i c .$
	Answered by mrW1 last updated on 22/Feb/17

	$A B C D i s a c o n v e x q u a d r i l a t e r a l .$ $E F G H i s t h e q u a d r i l a t e r a l w h i c h i s$ $f o r m e d b y t h e a n g l e b i s e c t o r s o f$ $A B C D .$ $∠ A + ∠ B + ∠ C + ∠ D = 360 °$ $∠ E F G = 180 ° - ∠ F A B - ∠ F B A = 180 ° - \frac{1}{2} ∠ A - \frac{1}{2} ∠ B$ $∠ E H B = 180 ° - ∠ H C D - ∠ H D C = 180 ° - \frac{1}{2} ∠ C - \frac{1}{2} ∠ D$ $∠ E F G + ∠ E H B = 180 ° - \frac{1}{2} ∠ A - \frac{1}{2} ∠ B + 180 ° - \frac{1}{2} ∠ C - \frac{1}{2} ∠ D$ $= 360 ° - \frac{1}{2} (∠ A + ∠ B + ∠ C + ∠ D) = 360 ° - \frac{1}{2} \times 360 ° = 180 °$ $i n t h e s a m e w a y$ $∠ F E H + ∠ F G H = 180 °$ $∵ E F G H i s c y c l i c q u a d r i l a t e r a l .$ $B T W : R F G H i s a l w a y s c y c l i c, e v e n i f$ $A B C D i s n o t c y c l i c!$
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