In triangle $ABC$, prove that $I{O}^2=R^2\left(3-2\cos A-2\cos B-2\cos C\right)$ where $O \& I$ are circum-centre and in-centre of triangle $ABC$ respectively and $R$ is circum-radius.

In a triangle $ABC$, prove that $I{P}^2=2r^2-4R^2\cos A\cos B\cos C$, where $P \& I$ are orthocentre and incentre of triangle $ABC$ and $r \& R$ are inradius and circumradius of given triangle.

In a triangle $ABC$, prove that $O{G}^2=R^2-\frac{1}{9}\left(a^2+b^2+c^2\right)$, where $a, b, c$ are length of sides of triangle opposite to angle $A, B \& C$, respectively and $O \& G$ are circum-centre and centroid of triangle and $R$ is circum radius of given triangle.