In a triangle $ABC$, if $I$ is the in-centre and $I_1,I_2$ and $I_3$ are the centres of the described circles, then prove that

$I{I}_1=a\sec \frac{A}{2}$

In a triangle $ABC$ if $H$ is the orthocentre and $I$ the In-centre then prove that

$I{H}^2=2r^2-4R^2\cos A\cos B\cos C$

Given an isosceles triangle with lateral side of length $b$, base angle $\alpha <\frac{\pi }{4};$$R$ is circumradius and $r$ is radius of incircle and $O$, $I$ the centres of the circumcircle and incircle respectively, then prove that

$R=\frac{1}{2}bcosec\alpha$