Prove that the quadrilateral formed by the bisectors of internal angles of a cyclic quadrilateral is also cyclic.

Asked by Topperlearning User | 4th Oct, 2017, 02:03: PM

Expert Answer:

 
begin mathsize 12px style In space the space given space figure space ABCD space is space straight a space cyclic space quadrilateral.
AH comma space BF comma space CF space and space DH space are space the space angle space bisectors space of space angle space straight A comma space angle space straight B comma space angle space straight C space and space angle space straight D.
To space prove space colon space EFGH space is space also space straight a space cyclic space quadrilateral.
Proof space colon space
angle space FEH space equals space angle space AEB space space space space left parenthesis straight i right parenthesis space space space space space space space space space space space left parenthesis Vertically space opposite space angles right parenthesis
angle space FGH space equals space angle space DGC space space space left parenthesis ii right parenthesis space space space space space space space space space space space left parenthesis Vertically space opposite space angles right parenthesis
angle space FEH space space plus space angle space FGH space space equals space angle space AEB space plus space space angle space DGC space space space space space space space left parenthesis Adding space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis space right parenthesis
Now comma space by space angle space sum space property space of space straight a space triangle comma space angle space AEB space equals space 180 degree space minus space open parentheses 1 half angle space straight A space plus space 1 half angle space straight B close parentheses
and space angle space DGC space equals space 180 degree space minus space space open parentheses 1 half angle space straight C space plus space 1 half angle space straight D close parentheses
rightwards double arrow angle space FEH space plus space angle space FGH space equals space 180 degree space minus space space open parentheses 1 half angle space straight A space plus space 1 half angle space straight B close parentheses space plus space 180 degree space minus space space open parentheses 1 half angle space straight C space plus space 1 half angle space straight D close parentheses
rightwards double arrow angle space FEH space plus space angle space FGH space equals space 360 degree space minus 1 half space space open parentheses angle space straight A space plus space angle space straight B space plus space angle space straight C space plus space angle space straight D close parentheses space
rightwards double arrow angle space FEH space plus space angle space FGH space equals space 360 degree space minus 1 half space cross times 360 degree space
rightwards double arrow angle space FEH space plus space angle space FGH space equals space 180 degree
Now comma space the space sum space of space opposite space angels space of space quadrilateral space EFGH space is space space 180 degree.
Therefore comma space EFGH space is space straight a space cyclic space quadrilateral.
space space space space space space space space space space space space space space space end style

Answered by  | 4th Oct, 2017, 04:03: PM