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.

Question

Aditya Gupta · Answer

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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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.

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