Prove that the perimeter of a right triangle is equal to the sum of the diameter of its incircle and twice the diameter of its circumcircle.

Asked by ashutoshtripathy2000 | 2nd Oct, 2015, 10:52: AM

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.
 
triangle A B C i s space a space r i g h t space t r i a n g l e space r i g h t space a n g l e d space a t space B. A n g l e space i n space a space s e m i space c i r c l o e space i s space 90 degree T h u s comma space A C space i s space t h e space d i a m t e r space o f space t h e space c i r c l e. O space i s space t h e space i n c e n t r e comma space apostrophe r apostrophe space i s space t h e space i n r a d i u s space o f space t h e space c i r c l e space a n d space B P space a n d space B Q space a r e space tan g e n t s. T a n g e n t s space f r o m space a n space e x t e r n a l space p o i n t space t o space a space c i r c l e space a r e space e q u a l. T h u s comma space B P equals B Q I n space a space s i m i l a r space m a n n e r comma space C Q equals C R space a n d space A P equals A R S i n c e space B P equals B Q space a n d space B P space a n d space B Q space a r e space t h e space tan g e n t s comma space s o comma space O Q B P space i s space a space s q u a r e. rightwards double arrow B P equals B Q equals O P equals O Q equals r N o w space c o n s i d e r comma space A B plus B C plus C A equals A P plus B P plus B Q plus C Q plus A C space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals A R plus O P plus O P plus C R plus A C space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals A R plus C R plus 2 O P plus A C space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals A C plus 2 O P plus A C space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 A C plus 2 r space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 A C plus d space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 open parentheses D i a m e t e r space o f space c i r c u m c i r c l e close parentheses plus open parentheses D i a m e t e r space o f space i n c i r c l e close parentheses T h u s comma space t h e space p e r i m e t e r space o f space a space r i g h t space t r i a n g l e space i s space e q u a l space t o space t h e space s u m space o f space t h e space d i a m e t e r space o f space i t s i n c i r c l e space a n d space t w i c e space t h e space d i a m e t e r space o f space i t s space c i c u m c i r c l e.

Answered by Vimala Ramamurthy | 2nd Oct, 2015, 05:16: PM

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