The following result was experimentally (re)discovered by myself using the dynamic geometry software, Sketchpad, which is available as a free download - please see the Link at the bottom of this page.
Theorem
The respective intersections E, F, G and H of the angle bisectors of angles A, B, C and D of a cyclic quadrilateral ABCD, with the circumcircle, form a rectangle.
(Drag any of the red vertices/points).
Cyclic Quadrilateral Angle Bisectors Rectangle Result
Challenge
1) Can you explain (prove) why EFGH is a rectangle? Can you explain (prove) why it is true in different ways?
Hint: The result follows almost directly from a useful Lemma proved on p. 190 of my book Some Adventures in Euclidean Geometry (free to download) or in printed bookform from Some Adventures in Euclidean Geometry (Print).
Explore More
2) Is the result also true if ABCD becomes a crossed quadrilateral?
Unfortunately in the sketch above when ABCD becomes a crossed quadrilateral, because of the way points are defined on the circle by the software, the rectangle degenerates to a line. So you'll need to construct your own dynamic geometry sketch to explore this case or view this dynamic sketch Cyclic Quadrilateral Angle Bisectors Rectangle with Cinderella. This software is also available as a free download at: The Interactive Geometry Software Cinderella.
Further Generalization
3) Can you apply or generalize the result to a cyclic hexagon? Click on the linked button 'Generalization to Cyclic Hexagon' in the sketch above.
Reference
De Villiers, M. (2011). An interesting cyclic quadrilateral result. Mathematical Digest, no. 163, April 2011, pp. 6-7.
Some Related Links
Logical Paradox (Rethinking Proof activity)
Cyclic Quadrilateral Incentres Rectangle
Triangle Circumcircle Incentre Result
Cyclic Quadrilateral Midpoints of Arcs Theorem
Cyclic Hexagon Alternate Angles Sum Theorem
A generalization of the Cyclic Quadrilateral Angle Sum theorem
Cyclic Quadrilateral Difference of Squares 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
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
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 17 January 2011, converted to WebSketchpad 25 Feb 2019; updated 5 May 2025; 13 Sept 2025.