Finding the length of the line segment $JI$

Question

In the diagram, $J$ is the circumcenter of $\Delta ABC$ and $I$ is the midpoint of $BC$.

How can i show that $JI=\cfrac {R}{ \cos A}$ ?

I simple don't know how to prove such simple fact...

Answer 1

I don't think that $JI=\frac{R}{\cos A}$ holds.

Consider the right triangle $JIC$ where we have $JC=R,IC=\frac{BC}2$ with $2R=\frac{BC}{\sin A}$ by the law of sines. Then, we have $$JI=\sqrt{R^2-\left(\frac{BC}{2}\right)^2}=\sqrt{R^2-R^2\sin^2 A}=R\sqrt{\cos^2A}=R|\cos A|.$$