Notes
divided hexagon iii solution

Divided Hexagon III

Solution by Similarity and Properties of a Regular Hexagon

With the points labelled as above, angles $D \hat{I} F$ and $H \hat{I} B$ are equal since they are vertically opposite and angles $F \hat{E} D$ and $B \hat{A} H$ are both interior angles in a regular hexagon so are equal. This means that the two kites are similar. As the area scale factor is $9$, the length scale factor is $3$.

The length of $A I$ is then a third of $E I$, so $I$ is three quarters of the way from $E$ to $A$ and so is level with $C G$. Also, the length of $A B$ is one third of that of $E F$, so triangle $A C I$ has area three times that of triangle $A B I$ and so triangle $A C G$ has area $3$.

The full hexagon therefore has area $18$. The blue and red areas consist of $10$ of this, leaving $8$ for the two side regions. The yellow region therefore has area $4$.