Notes
two congruent quadrilaterals and a rectangle solution

Two Congruent Quadrilaterals and a Rectangle

Solution by Triangle Area and Congruent Shapes

As the line segment $G C$ is common to both shapes, they are related by a rotation of $180^\circ$ about a point which can be found by joining two corresponding points, such as $A$ and $E$, and finding its midpoint, which is $I$. In particular, $A I$ and $I E$ are of equal length.

Since $B H$ is parallel to $A C$, the perpendicular distances of $B$ and $H$ abvoe $A C$ are equal, so triangles $A B C$ and $A H C$ have the same area. Then as $A I$ and $I E$ have the same length, triangles $A I H$ and $I E H$ have the same area. Putting these together shows that regions $A B C I$ and $I E H$ have the same area. Therefore the yellow and green regions have equal area.