What is the value of the segment $BI$ the figure below? For reference: In the triangle $ABC$ draws the bisector, $BD, ~D ~and ~AC$, which intersects the circumference circumscribed in $E$. If $AE = 4~ and~ BE=8$, calculate $BI$.($I$ ➔ incenter of triangle $ABC$)
My progress:
My drawing
$\measuredangle ABE = \measuredangle EBC = \measuredangle EAC = \measuredangle ACE = \theta \implies \measuredangle AOC = 4\theta\\
O = orthocenter$


Since $\measuredangle AIE=\measuredangle EAI,$ we obtain: $$BI=BE-IE=BE-AE=8-4=4.$$
