Sum of polygonal areas

The GOLD dot is connected to all four vertices of a rectangle. You can set the size & shape of the rectangles by dragging the GRAY dot. You can place the GOLD dot anywhere in the rectangle. The length of the segment from each vertex of the rectangle to the GOLD dot is the side of a regular polygon. The sum of the areas of the BLUE polygons is equal to the sum of the areas of the GREEN polygons no matter where you place the GOLD dot. Can you prove this? What happens when you drag the GOLD dot outside the rectangle? Why?