Prove that the quadrilateral formed by the internal angle bisectors of any quadrilateral is cyclic ?

Asked by shub3893 | 4th Dec, 2015, 09:04: PM

Expert Answer:

C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
L e t space A B C D space b e space t h e space g i v e n space q u a d r i l a t e r a l. therefore angle A plus angle B plus angle C plus angle D equals 360 degree D i v i d i n g space b y space 2 space i n space b o t h space t h e space s i d e s comma space w e space h a v e comma fraction numerator angle A plus angle B plus angle C plus angle D over denominator 2 end fraction equals fraction numerator 360 degree over denominator 2 end fraction rightwards double arrow fraction numerator angle A plus angle B plus angle C plus angle D over denominator 2 end fraction equals 180 degree.... left parenthesis 1 right parenthesis  D r a w space t h e space a n g l e space b i s e c t o r s space A P comma space B Q comma space C R space a n d space D S. T h u s comma space P Q R S space i s space a space q u a d r i l a t e r a l. W e space n e e d space t o space p r o v e space t h a t space P Q R S space i s space a space c y c l i c space q u a d r i l a t e r a l. T h a t space i s space t o space p r o v e space t h a t comma space angle P plus angle R equals 180 degree rightwards double arrow angle A P B plus angle C R D equals 180 degree  S i n c e space A P space i s space t h e space a n g l e space b i s e c t o r comma space w e space h a v e comma space angle P A B equals 1 half angle A... left parenthesis 2 right parenthesis S i n c e space B Q space i s space t h e space a n g l e space b i s e c t o r comma space w e space h a v e comma space angle P B A equals 1 half angle B... left parenthesis 3 right parenthesis C o n s i d e r space t h e space t r i a n g l e space A P B colon B y space a n g l e space s u m space p r o p e r t y comma space w e space h a v e comma angle P A B plus angle P B A plus angle A P B equals 180 degree... left parenthesis 4 right parenthesis  S i n c e space C R space i s space t h e space a n g l e space b i s e c t o r comma space w e space h a v e comma space angle R C D equals 1 half angle C... left parenthesis 5 right parenthesis S i n c e space D S space i s space t h e space a n g l e space b i s e c t o r comma space w e space h a v e comma space angle R D C equals 1 half angle D... left parenthesis 6 right parenthesis C o n s i d e r space t h e space t r i a n g l e space C R D colon B y space a n g l e space s u m space p r o p e r t y comma space w e space h a v e comma angle R C D plus angle R D C plus angle C R D equals 180 degree... left parenthesis 7 right parenthesis  A d d i n g space e q u a t i o n s space left parenthesis 4 right parenthesis space a n d space left parenthesis 7 right parenthesis comma space w e space h a v e comma angle P A B plus angle P B A plus angle A P B plus angle R C D plus angle R D C plus angle C R D equals 180 degree plus 180 degree R e w r i t i n g space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma angle P A B plus angle P B A plus angle R C D plus angle R D C plus angle A P B plus angle C R D equals 360 degree U sin g space e q u a t i o n s space left parenthesis 2 right parenthesis comma space left parenthesis 3 right parenthesis comma space left parenthesis 5 right parenthesis space a n d space left parenthesis 6 right parenthesis comma space i n space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma 1 half angle A plus 1 half angle B plus 1 half angle C plus 1 half angle D plus angle A P B plus angle C R D equals 360 degree rightwards double arrow 1 half open parentheses angle A plus angle B plus angle C plus angle D close parentheses plus angle A P B plus angle C R D equals 360 degree U sin g space e q u a t i o n space left parenthesis 1 right parenthesis space i n space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma rightwards double arrow 180 degree plus angle A P B plus angle C R D equals 360 degree rightwards double arrow angle A P B plus angle C R D equals 360 degree minus 180 degree rightwards double arrow angle A P B plus angle C R D equals 180 degree T h u s comma space w e space h a v e space p r o v e d space t h a t space t h e space s u m space o f space t h e space o p p o s i t e space a n g l e s space a r e space s u p p l e m e n t a r y. T h e r e f o r e comma space P Q R S space i s space a space c y c l i c space q u a d r i l a t e r a l.

Answered by Vimala Ramamurthy | 5th Dec, 2015, 02:29: PM