Question 5
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ,$ $\angle ABC = 60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D,$ $E,$ and $F$ be the midpoints of sides $\overline{BC},$ $\overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $\triangle DEF$ intersects $\overline{BD},$ $\overline{AE},$ and $\overline{AF}$ at points $G,$ $H,$ and $J,$ respectively. The points $G,$ $D,$ $E,$ $H,$ $J,$ and $F$ divide the circumcircle of $\triangle DEF$ into six minor arcs, as shown. Find $\overset{\frown}{DE} + 2 \cdot \overset{\frown}{HJ} + 3 \cdot \overset{\frown}{FG},$ where the arcs are measured in degrees.
