SE-SKOLORNAS 2015 Problem 1

Given the acute triangle $ABC$ . A diameter of the circumscribed circle of the triangle intersects the sides $AC$ and $BC$ , dividing the side $BC$ in half. Show that the same diameter divides the side $AC$ in a ratio of $1: 3$ , calculated from $A$ , if and only if $\tan B = 2 \tan C$ .