Show that the center of the incircle of a triangle is the point of intersection of the angle bisectors.

Asked by araima2001 | 3rd Nov, 2015, 09:51: 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 O space b e space t h e space c e n t r e space o f space t h e space i n c i r c l e space a space t r i a n g l e. T h u s comma space O D space a n d space O E space b e space t h e space i n r a d i u s space o f space t h e space i n c i r c l e. rightwards double arrow O D equals O E C o n s i d e r space t h e space t r i a n g l e s comma space triangle B D O space a n d space triangle B E O B D space a n d space B E space a r e space t h e space tan g e n t s space t o space t h e space i n c i r c l e. A n d space h e n c e comma space angle B D O equals angle B E O equals 90 degree B O equals B O space open square brackets c o m m o n close square brackets S i n c e space tan g e n t s space f r o m space a space p o i n t space a r e space e q u a l comma space w e space h a v e comma space B D equals B F T h u s comma space b y space R H S space c o n g r u e n c e comma space w e space h a v e comma space triangle B D O asymptotically equal to triangle B E O T h e space c o r r e s p o n d i n g space p a r t s space o f space t h e space c o n g r u e n t space t r i a n g l e s space a r e space c o n g r u e n t. T h e r e f o r e comma space angle F B O equals angle O B D T h a t space i s space O B space i s space t h e space a n g l e space b i s e c t o r space o f space t h e space a n g l e space angle F B D. S i m i l a r l y space w e space c a n space p r o v e space t h a t comma space O A space i s space t h e space a n g l e space b i s e c t o r space o f space angle F A C A n d space O C space i s space t h e space a n g l e space b i s e c t o r space o f space t h e space a n g l e space angle A C B. T h u s space a l l space t h e space a n g l e space b i s e c o t r s space i n t e r s e c t space a t space t h e space p o i n t space O. T h a t space i s space t h e space a n g l e space b i s e c o t r s space i n t e r s e c t space a t space t h e space i n c e n t r e space o f space t h e space i n c i r c l e.

Answered by Vimala Ramamurthy | 4th Nov, 2015, 12:08: PM

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