Crossed Quadrilateral PropertiesNOTE: Please WAIT while the applets below load. If a security pop-up menu appears in your browser, please choose RUN/ALLOW to let the applets run properly. It is completely safe & can be trusted. If you have the very latest Java on your PC or Apple Mac, and experience problems with the applets loading, please go here for additional information on Java settings that should resolve the issue. A crossed or bowtie quadrilateral ABCD is shown below. In other words, it is a quadrilateral with a pair of opposite sides crossing each other. Angle Sum 1) Consider the marked angles, and their displayed "anglesum =" in the menu on the left. Drag any of the vertices to change the shape of the crossed quadrilateral dynamically. What you do notice? Can you write down a conjecture? 2) Can you EXPLAIN WHY (prove that) your conjecture is true?
For some hints and more info go to General formula for interior angle sum Michael de Villiers, Originally created Jan 2011 with GeoGebra. |
Sum of opposite angles when cyclic
1) What do you notice about the two sums of the marked angles at opposite vertices C and E, and D and F of the crossed cyclic quadrilateral CDEF? Note that the calculations ε = α + γ and ζ = β + δ are displayed in the left column. Drag any of the vertices to check.
2) Can you EXPLAIN WHY (prove that) your conjecture is true?
As mentioned in my book Some Adventures in Euclidean Geometry, we can in general say that for cyclic quadrilaterals, both convex and crossed, that the two sums of opposite angles are equal (180o when convex and 360o when crossed).
Michael de Villiers, Created 28 Sept 2014 with GeoGebra.
Varignon's Theorem
Varignon's theorem is named after the French mathematician and engineer Pierre Varignon in 1731. The theorem states that the midpoints of the sides of a quadrilateral form a parallelogram, and is also true for a crossed qadrilateral as shown below. (Prove it!)
1) An interesting additional property of the theorem is that the Area Ratio between ABCD and EFGH is always 2, as also shown below for the crossed quadrilateral. (See the measurements in the sketch and the calculation "AREARATIO =" in the left column). Drag any of the vertices to explore and convince yourself that this is the case.
2) Can you figure out and EXPLAIN HOW the area of the crossed quadrilateral is determined and defined by GeoGebra, and hence WHY (prove that) the area ratio result is also true for crossed quadrilaterals?
A discussion of how to determine the area of a crossed quadrilateral is given on pp. 177-178 of my book Rethinking Proof with Sketchpad and in the classic book by Coxeter & Greitzer Geometry Revisited. Alternatively, read my 2014/2015 paper Slaying a Geometrical Monster: Finding the Area of a Crossed Quadrilateral.
Further related readings:
1) The role and function of quasi-empirical methods in mathematics Canadian Journal of Science, Mathematics and Technology Education, July 2004, pp. 397-418.
2) I have a dream ... Crossed Quadrilaterals: A Missed Lakatosian Opportunity? Philosophy of Mathematics Education Journal, July 2015.
(This bit of prose in the YouTube video above is intended as a critical satire of the still dominant practice in mathematics education to exclude crossed quadrilaterals from the set of quadrilaterals, and hence from the curriculum).
Michael de Villiers, Created 28 Sept 2014 with GeoGebra.