Another parallelogram area ratio

In my Rethinking Proof book the following problem was given in relation to the dynamic figure below:
"Determine the area of the shaded quadrilateral as a fraction of the area of the parallelogram, where the boundaries of the quadrilateral are lines drawn from the vertices to the midpoints of sides as shown."

1) Click on the 'Show Areas' button below to view the ratio between the two areas concerned. Drag any of the red vertices to explore.
2) What sort of quadrilateral is the shaded quadrilateral? Why? Can you drag the parallelogram so that the shaded quadrilateral becomes a rectangle, rhombus or square?


Another parallelogram area ratio

3) Can you explain why (prove that) the area ratio observed above is invariant (constant)? Can you explain (prove) it in more than one way?
Click on the 'Half turn triangles' button. What do you notice? Use this observation to explain (prove) the area ratio.

Generalize Further
4) What happens if ABCD is not a parallelogram, but a general convex quadrilateral?
5) If the same construction is repeated to obtain a shaded quadrilateral, does the area ratio remain the same? Explore with a dynamic construction of your own.
6) Check your answer in 5) above by going to Sylvie's Theorem.

De Villiers, M. (1999/2003/2012). Rethinking Proof. McGraw-Hill, pp. 74-75.

Related Links
International Mathematical Talent Search (IMTS) Problem Generalized
An Area Preserving Transformation: Shearing
Sylvie's Theorem
Some Parallelo-hexagon Area Ratios
Area Parallelogram Partition Theorem: Another Example of the Discovery Function of Proof
Finding the Area of a Crossed Quadrilateral

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Created by Michael de Villiers with WebSketchpad, 4 Oct 2023.