The result below appeared on the cover of the Learning & Teaching Mathematics journal in June 2008.
Theorem
A cyclic quadrilateral defines four arcs. Then the lines joining the midpoints of opposite arcs are perpendicular.
Cyclic Quadrilateral Midpoints of Arcs Theorem
Challenge
Can you explain why (prove that) the result is true?
Apart from the Hint in the sketch, have a look at this Lemma: Further Hint about Arc Angles.
If the above hints don't help: A proof of this result can be found on p. 190 of Some Adventures in Euclidean Geometry (free to download) or in printed bookform from Some Adventures in Euclidean Geometry (Print).
NOTE: Due to the way the Sketchpad sketch was constructed & how some constructions are defined in the software, the above sketch might suddenly show the two lines parallel when the cyclic quadrilateral is dragged until it becomes "crossed". Have a look at this Cinderella sketch: Cyclic Quad Arc Result that doesn't show this "discontinuity". (Drag any of A, B, C or D in the sketch). Cinderella can also be downloaded for free at: Download Cinderella.
Application
This result is useful in proving this result: Cyclic Quadrilateral Incentres Rectangle.
Reference
De Villiers, M. (2009). LTM Cover Problem, June 2008. Learning & Teaching Mathematics, July, p. 57.
Some Related Links
Cyclic Quadrilateral Incentres Rectangle
Triangle Circumcircle Incentre Result
Cyclic Quadrilateral Angle Bisectors Rectangle Result
Cyclic Hexagon Alternate Angles Sum Theorem
A generalization of the Cyclic Quadrilateral Angle Sum theorem
Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi
Alternate sides cyclic-2n-gons and Alternate angles circum-2n-gons: Generalizations of isosceles trapezia and kites
Angle Divider Theorem for a Cyclic Quadrilateral
Conway’s Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Some Circle Concurrency Theorems
Converse of Tangent-Secant Theorem (Euclid Book III, Proposition 36)
Some Generalizations of Napoleon's Theorem
A Cyclic Quadrilateral Generalization of a Golden Rectangle property
A variation of Miquel's theorem and its generalization
Minimum Area of Miquel Circle Centres Triangle
Matric Exam Geometry Problem - 1949
A 1999 British Mathematics Olympiad Problem involving a Cyclic Hexagon
An extension of the IMO 2014 Problem 4
Six Point Cevian Circle
Nine Point Conic and Generalization of Euler Line
Geometry Loci Doodling with Cyclic Quadrilaterals
Crossed Quadrilateral Properties
Cyclic Kepler Quadrilateral Conjectures
Cross's (Vecten's) theorem & generalizations to quadrilaterals
Twin Circles for a Van Aubel configuration involving Similar Parallelograms
Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals
Nine-point centre (anticentre or Euler centre) & Maltitudes of Cyclic Quadrilateral
Cyclic Quadrilateral Difference of Squares Theorem
Some further generalizations of an associated result of the Van Aubel configuration using pairs of similar triangles
Eight Point Conic for Cyclic Quadrilateral
Bradley's Theorem for a Circumscribed Quadrilateral
Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula
The quasi-circumcentre and quasi-incentre of a quadrilateral
Bicentric Quadrilateral Properties
Concurrent Angle Bisectors of a Quadrilateral
Visually Introducing & Classifying Quadrilaterals by Dragging (Grades 1-7)
An Inclusive, Hierarchical Classification of Quadrilaterals
Some External Links
Incenters in Cyclic Quadrilateral (Cut The Knot)
Cyclic quadrilateral (Wikipedia)
Cyclic Quadrilateral (Wolfram MathWorld)
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 JavaSketchpad, Dec 2009: Updated 23 July 2012; updated to WebSketchpad, 5/6 May 2025.