Note: the following curious, paradoxical geometry problem is yet another variation of an International Mathematical Talent Search (IMTS) problem available here.
"Determine the area of the crossed octagon ALBMCNDO as a fraction of the area of the square ABCD, where L, M, N, O are the midpoints of the sides of the square as indicated."
1) Click on the 'Show Areas' button below to view the two areas concerned. Drag any of the red vertices to explore.
2) What do you notice about the area of the crossed octagon ALBMCNDO? Do you find it surprising? Or perhaps even contradictory or confusing?
a) What is happening here? The situation looks quite paradoxical! b) So the question is WHY does the software show the area of the crossed octagon equal to that of the square while the total area shaded in green is clearly less than that of the square? Can the situation be explained & clarified?
3) Also note that the formed crossed octagon is not regular. However, it has all sides equal and alternate angles are equal as can be seen by clicking on the 'Show Side Lengths' and 'Show Angles' buttons. This crossed octagon is therefore what I've called a semi-regular side-gon, specifically in this case, a (crossed) 'semi-regular side-octagon' - which is a generalization to 2n-gons of the concept of a rhombus.
Area of crossed semi-regular side-octagon
4) Paradox: Why is the area of the crossed octagon ALBMCNDO as calculated/determined by the software NOT equal to the total area of all the parts shown & shaded in 'green'by the software?
5) In order to correctly determine the area of a crossed polygon one has to use 'signed areas' as explained by Coxeter & Greitzer (1967). Using signed areas, can you explain why (prove that) the area of the crossed octagon ALBMCNDO equals the area of square ABCD as observed above? Can you explain (prove) it in more than one way?
A paper on this topic entitled "A Geometry Puzzler and its Explanation" will in due course be submitted to the Learning & Teaching Mathematics journal.
Coxeter, H. S. M. & Greitzer, S. L. (1967). Geometry Revisited. MAA, Washington, pp. 51-54.
De Villiers, M. (1994/2009). Some Adventures in Euclidean Geometry. Morrisville, NC: Lulu Publishers.
De Villiers, M. (1998). A Sketchpad discovery involving triangles and quadrilaterals. KZN Mathematics Journal, 3(1), 18-21.
De Villiers, M. (2014). Slaying a Geometrical Monster: Finding the Area of a Crossed Quadrilateral. Scottish Mathematical Council Journal, 44 (Dec), pp. 71-74.
De Villiers, M. (2015). I have a dream: Crossed Quadrilaterals - a Missed Lakatosian Opportunity?. Philosophy of Mathematics Education Journal, No. 29, July.
De Villiers, M. (2020). The Value of using Signed Quantities in Geometry. Learning and Teaching Mathematics, No. 29, pp. 30-34.
International Mathematical Talent Search (IMTS) Problem Generalized
Another parallelogram area ratio
Area ratios of some polygons inscribed in quadrilaterals and triangles
An Area Preserving Transformation: Shearing
Some Parallelo-hexagon Area Ratios
Area Parallelogram Partition Theorem: Another Example of the Discovery Function of Proof
Finding the Area of a Crossed Quadrilateral
Crossed Quadrilateral Properties
Created by Michael de Villiers with WebSketchpad, 9 Oct 2023.