JEE Advance - Mathematics (1999 - No. 36)
Let $$ABC$$ be a triangle having $$O$$ and $$I$$ as its circumcenter and in centre respectively. If $$R$$ and $$r$$ are the circumradius and the inradius, respectively, then prove that $${\left( {IO} \right)^2} = {R^2} - 2{\mathop{\rm Rr}\nolimits} $$. Further show that the triangle BIO is a right-angled triangle if and only if $$b$$ is arithmetic mean of $$a$$ and $$c$$.
The distance between the incenter and circumcenter is equal to the circumradius minus twice the inradius.
The distance between the incenter and circumcenter squared is equal to the circumradius squared minus twice the product of the circumradius and inradius.
The distance between the incenter and circumcenter is equal to the circumradius plus twice the inradius.
The distance between the incenter and circumcenter squared is equal to the circumradius squared plus twice the product of the circumradius and inradius.
The distance between the incenter and circumcenter squared is equal to the inradius squared minus twice the product of the circumradius and inradius.