Outer-Inner Polygon Area Ratios

 For a triangle, Routh's Theorem gives the ratio of the areas of the outer to inner triangles formed by lines joining each vertex to points on the opposite edge.

​In the case shown here, in which the sides are divided in the ratio pf 2 : 1 cyclically, the ratio os 7 : 1

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For a quadrilateral (depicted below), in the case of joining each vertex to the midpoint of the next but one edge (all done clockwise or all done counter-clockwise) it is easy to be misled by rounding errors into believing that the ratio of the areas of the outer to inner quadrilaterals is 5. In fact it lies between 5 and 6.

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​There is an applet for displaying the diagrams for this paper and for assisting in the exploration can be found here.