As you may already know, or have perhaps seen in the Parallelogram Angle Bisectors activity: the angle bisectors of the angles of a parallelogram form a rectangle.
What happens if?
1) But what happens if instead of angle bisectors, we construct the perpendicular bisectors of the sides of a parallelogram?
2) What sort of figure is formed? And what is the relationsship, if any, between the formed figure and the original?
Explore
3) Use the dynamic sketch below to explore this question by dragging.
4) Click on the 'Show Measurements' button to have a look at corresponding sides and angles.
5) What do you notice? Can you formulate a conjecture?
Perpendicular Bisectors of a Parallelogram
Challenge
You should have noticed that the formed quadrilateral D'C'B'A' is a parallelogram similar to ABCD.
6) Can you prove the result? What is the similarity factor? Can you prove it in more than one way?
(Compare your proof with the one given in Humenberger & De Villiers, 2026b).
7) Is it possible for D'C'B'A' to be congruent to ABCD? If so, when? If not, why not?
Explore More
8) What if we construct the perpendicular bisectors of a trapezoid? What figure is formed and what is the relationship, if any, with the original trapezoid?
9) Click on the 'Link to Trapezoid' button to navigate to a new sketch showing a trapezoid ABCD and its corresponding perpendicular bisector (PB) quadrilateral.
10) Explore the new figure by dragging and clicking on 'Show Measurements' button. What do you notice? Do your observations still hold if the trapezoid ABCD becomes crossed?
Challenge
11) Can you prove your observation in 10) above?
(Compare your proof with the one given in Humenberger & De Villiers, 2026a).
Further Exploration
12) What happens if we construct the perpendicular bisectors of a hexagon with opposite sides parallel?
13) Go to Perpendicular bisectors of a hexagon with opposite sides parallel to explore Q12.
14) What happens if we construct the perpendicular bisectors of a tangential quadrilateral?
15) Go to Perpendicular bisectors of a tangential quadrilateral to explore Q14.
References
Humenberger, H. & De Villiers, M. (2026a). Perpendicular bisector quadrilaterals as a substantial learning environment. Mathematics in School, 55, 1, 28-31.
Humenberger, H. & De Villiers, M. (2026b). Perpendicular Bisector Quadrilaterals of Parallelograms. Mathematics in School, 55, Z, XX-YY.
Related Links
Water Supply: Four Towns (Introduction to perpendicular bisectors - Rethinking Proof activity)
Water Supply: Three Towns (Concurrency of perpendicular bisectors - Rethinking Proof activity)
Cyclic Quadrilateral (Rethinking Proof activity)
Some Trapezoid (Trapezium) Explorations (See Investigation 5 about perpendicular bisectors)
The Perpendicular Bisectors of an Apollonius Quadrilateral
Parallelogram Angle Bisectors (Rethinking Proof activity)
IMO 2014 Problem 4 - Geometry
An extension of the IMO 2014 Problem 4
The quasi-circumcentre of a quadrilateral
The quasi-Euler line of a quadrilateral and of a hexagon
A theorem involving the perpendicular bisectors of a hexagon with opposite sides parallel
Perpendicular Bisectors of Circumscribed/Tangential Quadrilateral
Conway’s Circle Theorem as special case of Windscreen Wiper Theorem
A 1999 British Mathematics Olympiad Problem and its dual
The Equi-inclined Bisectors of a Cyclic Quadrilateral
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral
External Links
Spiral similarity (Wikipedia)
Perpendicular bisector construction of a quadrilateral (Wikipedia)
Quadrilaterals Formed by Perpendicular Bisectors (Cut The Knot)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Michael de Villiers, created with WebSketchpad, 12 Nov 2025; updated 23 Feb 2026.