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In any ∆ABC,if I is the centre of the inscribed Circle, and if AI is produced to meet the circumscribed Circle at O; show that O is the centre of the Circle circumscribed about the triangle BIC.

In any ∆ABC,if I is the centre of the inscribed Circle, and if AI is produced to meet the circumscribed Circle at O; show that O is the centre of the Circle circumscribed about the triangle BIC.

Grade:12th pass

1 Answers

Aditya Gupta
2075 Points
one year ago
note that angle AOC= angleIOC= angle B (angles on the same chord)
we know that incenter is the point of intersection of angle bisectors of a triangle.
now, angle OIC= angle IAC + angle ICA= A/2 + C/2.
so, angle OCI= 180 – angle IOC – angle OIC= 180 – B – (A/2 + C/2)
= A+C – (A/2 + C/2)= A/2 + C/2
hence angle OCI= angle OIC
so that OI= OC
similarly OI= OB
since OI= OB= OC= R (say), O is the centre of the Circle circumscribed about the triangle BIC.
kindly approve :)

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