I did not get directed angles and did not use the hint. This is my approach:
In the picture circle c$c$ is on points A$A$, E$E$ and F$F$.Circle d Circle $d$ is on points E$E$, F$F$, B$B$ and C$C$. GE$GE$ is tangent to circle d$d$ at E$E$, since ED$ED$ is a diameter of circle d$d$, then we have $GE\bot EM$ hencMEhence $ME$ is tangent to the circle c$c$.Now Now triangles GEM$\triangle GEM$ and GFM$\triangle GFM$ are equalcongruent due to SSS$SSS$.So So we have :
$\angle GFM=\angle GEM=90^o$$$\angle GFM=\angle GEM=90^{\circ}$$
This means that MF$MF$ is also tangent to circle c$c$. Now for line $IJ||BC$ ,$IJ\parallel BC$,H $H$ is the orthocenter of triangle ABC$\triangle ABC$, so AH$AH$ is perpendicular to BC$BC$. AH$AH$ is also the diameter of circle c$c$. since $IJ||BC$$IJ\parallel BC$ then $AH\bot IJ$$AH\perp IJ$, this means that IJ$IJ$ is tangent to circle c$c$ at point A$A$.
