A crossed or bowtie quadrilateral ABCD is shown below. In other words, it is a quadrilateral with a pair of opposite or alternate sides crossing each other.
Angle Sum
1) Consider the marked angles, and their displayed Angle Sum. Drag any of the vertices to change the shape of the crossed quadrilateral dynamically. What you do notice? Can you write down a conjecture for a crossed quadrilateral?
2) Can you EXPLAIN WHY (prove that) your conjecture for a crossed quadrilateral is true?
Crossed Quadrilateral Properties
Hint: For some hints and more info go to
Sum of opposite/alternate angles when cyclic
1) Click on the 'Link to cyclic alternate angles sum' button to navigate to a new sketch.
2) What do you notice about the two sums of the marked angles at opposite/alternate vertices P and R, and Q and S of the crossed cyclic quadrilateral PQRS? Drag any of the vertices to check.
3) Can you EXPLAIN WHY (prove that) your conjecture for a cyclic crossed quadrilateral is true?
Comment: As mentioned in my book Some Adventures in Euclidean Geometry, we can therefore in general say that for cyclic quadrilaterals, both convex and crossed, that the two sums of opposite/alternate angles are equal (to 180o when convex and 360o when crossed). Also note that the result about the sums of opposite/alternate angles for the crossed quadrilateral is equivalent to the geometry theorem usually stated as 'angles inscribed on the same chord of a circle are equal'.
Varignon's Theorem
1) Click on the 'Link to Varignon area' button to navigate to a new sketch.
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. (Prove it!)
2) An interesting additional property of the theorem is that the Area Ratio between PQRS and EFGH is always 2, as also shown for the crossed quadrilateral. Drag any of the vertices to explore and convince yourself that this is the case.
3) Can you figure out and EXPLAIN HOW the area of the crossed quadrilateral is determined and defined by Sketchpad, and hence WHY (prove that) the area ratio result is also true for crossed quadrilaterals?
Published Paper & Books
1) Read my paper Slaying a Geometrical Monster: Finding the Area of a Crossed Quadrilateral in the Scottish Mathematical Council Journal (2014) and in Learning and Teaching Mathematics (2015).
2) A discussion of how to determine the area of a crossed quadrilateral is also given on pp. 177-178 of my book Rethinking Proof with Sketchpad and in the classic book by Coxeter & Greitzer Geometry Revisited.
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.
3) The Value of using Signed Quantities in Geometry Learning & Teaching Mathematics, Dec 2020, no. 29.
(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).
Related Links
Interior angle sum of polygons (incl. crossed): a general formula
Investigating a general formula for the interior angle sum of polygons (A suggested guided learning activity starting with LOGO (turtle) geometry).
Cyclic Hexagon Alternate Angles Sum Theorem
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Created by Michael de Villiers, 2011; converted 27 June 2023 to WebSketchpad.