Bicentric Quadrilateral
A cyclic quadrilateral ABCD is circumscribed around a circle. (In general such quadrilaterals are called bicentric quadrilaterals).
Explore
1) Click on the 'Shown lines EG and FH' button.
2) What do you notice about the lines EG and FH joining the opposite tangential points?
3) Drag any of F or H to check your observation in 2).
4) Click on the 'Animate Point H' button for an animation.
Some Bicentric Quadrilateral Properties
Challenge
5) Can you explain why (prove that) the lines EG and FH are orthogonal (perpendicular) to each other?
Hint: If stuck, see problem 2 in Solutions to Reader Investigations: Feb 2003 from the AMESA KZN Mathematics Journal.
(Other Reader Investigations are available at Reader Investigations: 2000-2005 (but still needs to be updated).
Bicentric quadrilaterals have many interesting properties, and in recent years there has been some renewed interest, as shown in the references below, to explore their properties further. Below are some more properties that are quite a bit more challenging to prove than the one given above.
More Properties
6) Click on the'Show More Properties' button to show more properties.
7) What do you notice? Check by dragging or using the animation button.
Challenge
8) Can you explain why (prove that) the properties in 6) or 7) are true?
Hints
a) The property that I, O and the intersection of EG and FH are collinear, follows directly from this Lemma.
b) The property that the diagonals of ABCD are concurrent with the intersection of EG and FH, is probably most easily proved with projective geometry (e.g. see Aref & Wernick, 1968, p. 209.)
Special Cases
9) Can you drag ABCD into the shape of other quadrilaterals? Specifically, can you drag it into the shape of a Kite, Isosceles Trapezoid or Square?
Note: The cover of De Villiers (1994; 2009), shows a bicentric isosceles trapezoid as well as a bicentric kite - see below.
Constructing a general Bicentric Quadrilateral
A simple example of a bicentric quadrilateral is a square. But can you find a method to accurately construct a (general) bicentric quadrilateral by hand or by using dynamic geometry software? If so, can you find more than one method?
11) Check or compare your solutions with those at: Some other bicentric quadrilateral constructions.
References
Aref, M.N. & Wernick, W. (1968). Problem & Solutions in Euclidean Geometry. Dover Publications: New York, p. 209 (excerpt).
Bataille, M. (2009). A Duality for Bicentric Quadrilaterals. Crux Mathematicorum (with Mathematical Mayhem), Vo. 35, Issue 5, pp. 310-312.
De Villiers, M.D. (1994; 2009). Some Adventures in Euclidean Geometry (free to download as PDF). USA: Lulu Publishers.
(It is also available in printed bookform from Some Adventures in Euclidean Geometry (Print)).
Humenberger, H. (2023). On Six Collinear Points in Bicentric Quadrilaterals. Mathematics Magazine, 96:3, 285-295.
Josefsson, M. (2010). Characterizations of Bicentric Quadrilaterals. Forum Geometricorum, Volume 10, 165–173.
Josefsson, M. (2011). The area of a bicentric quadrilateral. Forum Geometricorum, 11: 155–164.
Josefsson, M. (2012). Maximal area of a bicentric quadrilateral. Forum Geometricorum, 12: 237–241.
Pillay, P. (2025). Some Notes on Bicentric Quadrilaterals. (Lecture notes for use by mathematical olympiad contestants).
Some Related Links
Constructing a general Bicentric Quadrilateral
Some other bicentric quadrilateral constructions
Some Properties of Bicentric Isosceles Trapezia & Kites
Triangle Circumcircle Incentre Result
Cyclic Quadrilateral Difference of Squares Theorem
Japanese theorem for cyclic quadrilaterals
Cyclic Quadrilateral Angle Bisectors Rectangle Result
Quadrilateral Similar Triangles Collinearity
Cyclic Hexagon Alternate Angles Sum Theorem
Circumscribed Hexagon Alternate Sides Theorem
Pitot's Theorem for a tangential quadrilateral
Converse of Pitot's theorem for a tangential quadrilateral
Angle Divider Theorem for a Cyclic Quadrilateral
Side Divider Theorem for a Circumscribed/Tangential Quadrilateral
Conway’s Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Tangential Quadrilateral Theorem of Gusić & Mladinić
Perpendicular Bisectors of Tangential Quadrilateral Theorem
Concurrency, collinearity and other properties of a particular hexagon
An interesting collinearity
More Properties of a Bisect-diagonal Quadrilateral
Euler-Nagel line Duality (analogy)
Concurrent Angle Bisectors of a Quadrilateral
Visually Introducing & Classifying Quadrilaterals by Dragging (Grades 1-7)
An Inclusive, Hierarchical Classification of Quadrilaterals
Some External Links
Bicentric quadrilateral (Wikipedia)
Right kite (Wikipedia)
Isosceles tangential trapezoid (Wikipedia)
Bicentric Quadrilateral (Wolfram MathWorld)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers as well as books on problem solving 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, Feb 2003; updated 7 April 2011; Updated to WebSketchpad, 16 May 2025.