Another parallelogram area ratio

Problem
In my Rethinking Proof with Geometer's Sketchpad book (free to download) 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."

Explore
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

Challenge
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 Areas Activity from Rethinking Proof or Sylvie's Theorem.

Reference
De Villiers, M. (1999/2003/2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press, pp. 74-75.

Related Links
Varignon Area (Rethinking Proof Activity)
Feynman's Triangle: Some Generalizations & Variations
Feynman Parallelogram Area Ratio Generalization
International Mathematical Talent Search (IMTS) Problem Generalized
A Geometric Paradox Explained (Another variation of an IMTS problem)
An Area Preserving Transformation: Shearing
Sylvie's Theorem
Areas Activity from Rethinking Proof
Some Parallelo-hexagon Area Ratios
Area Parallelogram Partition Theorem: Another Example of the Discovery Function of Proof
Finding the Area of a Crossed Quadrilateral (PDF)
Area Formula for Quadrilateral in terms of its Diagonals
Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula
Maximum area of quadrilateral problem
Minimum Area of Miquel Circle Centres Triangle
Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem
The Equi-partitioning Point of a Quadrilateral

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Created by Michael de Villiers with WebSketchpad, 4 Oct 2023; updated 11 July 2025.