Below are given two dynamic, interactive sketches of a bicentric isosceles trapezium as well as a bicentric kite. Definitions are given below. If not familar with the following properties, it is left to the reader to show that an isosceles trapezium is cyclic and that a kite has an incircle.
| Isosceles Trapezium | Kite |
| Any quadrilateral with at least one axis of symmetry through a pair of opposite sides. | Any quadrilateral with at least one axis of symmetry through a pair of opposite 'angles' (vertices). |
| Bicentric Isosceles Trapezium | Bicentric Kite |
| Any Isosceles Trapezium with an incircle. | Any Kite with a circumcircle (i.e. it is cyclic). |
Investigation 1: Bicentric Isosceles Trapezium
The bicentric isosceles trapezium is also called a 'circumscribed isosceles trapezium' - the term I use in my book Some Adventures in Euclidean Geometry (free to download) - as well as an 'isosceles tangential trapezoid' (Wikipedia; Josefsson, 2010).
1) Click on the 'Show Angles' button. What do you notice? Can you prove your observation?
2a) Can you find a formula that gives the perimeter of ABCD in terms of the area of ABCD and ∠ADC?
2b) Click on the 'Show Perimeter 1' button to check your answer in 2a).
3a) Can you find a formula for the area of ABCD in terms of just sides a and b?
3b) Click on the 'Show Area' button to check your answer in 3a).
4a) Can you find a formula for the perimeter of ABCD in terms of just sides a and b?
4b) Click on the 'Show Perimeter 2' button to check your answer in 4a).
5a) Can you find a formula for the distance between the incentre and incentre, IO, in terms of sides a and b?
5b) Click on the 'Show IO' button to check your answer in 5a).
6a) Can you find a formula for the radius, r, of the incircle, in terms of sides a and b?
6b) Click on the 'Show r' button to check your answer in 6a).
7a) Can you find a formula for the radius, R, of the circumcircle, in terms of sides a and b?
7b) Click on the 'Show R' button to check your answer in 7a).
Some Properties of Bicentric Isosceles Trapezia & Bicentric Kites
Investigation 2: Bicentric or Right Kite
The bicentric kite has also been called a 'right kite' in my book Some Adventures in Euclidean Geometry, which is available as a free download at the preceding link.
1) Click on the 'Link to bicentric kite' button to navigate to the next sketch.
2a) Can you find the perimeter of ABCD in terms of sides a and b?
2b) Click on the 'Show Perimeter' button to check your answer in 2a).
3a) Can you find a formula for the area of ABCD in terms of sides a and b?
3b) Click on the 'Show Area' button to check your answer in 3a).
4a) Can you find a formula for the radius, r, of the incircle, in terms of sides a and b?
4b) Click on the 'Show r' button to check your answer in 4a).
5a) Can you find a formula for the radius, R, of the circumcircle, in terms of sides a and b?
5b) Click on the 'Show R' button to check your answer in 5a).
6a) Can you find a formula for the distance between the incentre and incentre, IO, in terms of sides a and b?
6b) Click on the 'Show IO' button to check your answer in 6a).
Explore Generalizations
1) Can you generalize the above results further to other polygons? Can you generalize in more than one way?
2) Make conjectures and use dynamic geometry to explore further!
References
De Villiers, M. (1994; 2009). Some Adventures in Euclidean Geometry (free to download). USA: Lulu Publishers.
De Villiers, M. (2019). More Area, Perimeter and Other Properties of Circumscribed Isosceles Trapeziums and Cyclic Kites. Learning & Teaching Mathematics, no. 27, pp. 30-33.
Josefsson, M. (2010). Characterizations of Bicentric Quadrilaterals. Forum Geom., 10, pp. 165-173.
Rajpoot, H.C. (2022). Mathematical Analysis of Circum-inscribed Polygons: Circum-inscribed (C-I) Trapezium. Indian Institute of Technology Bombay.
Some Related Links
Bicentric Quadrilateral Properties
Constructing a general Bicentric Quadrilateral
Some other bicentric constructions
Bicentric Quadrilateral Area Formula in terms of angles, r & R (Click on link in sketch)
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
Opposite Side Quadrilateral Properties by Kalogerakis
Another Property of an Opposite Side Quadrilateral
A Property of an Opposite Angle Quadrilateral
Angle Bisectors form an Isosceles Trapezium (click on the 'Link to Right Pythagorean quad2' button)
Some Trapezoid (Trapezium) Explorations
Pitot's Theorem for a tangential or circumscribed quadrilateral
Theorem of Gusić & Mladinić
Japanese Circumscribed Quadrilateral Theorem
Perpendicular-Bisectors of Tangential Quadrilateral Theorem
Cyclic Kepler Quadrilateral Conjectures
Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals
Bradley's Theorem for a Circumscribed Quadrilateral
A generalization of the Cyclic Quadrilateral Angle Sum theorem
The Tangential (or Circumscribed) Polygon Alternate Sides Sum theorem
Area Formula for Quadrilateral in terms of its Diagonals
Concurrent Angle Bisectors of a Quadrilateral
Side Divider Theorem for a Circumscribed/Tangential Quadrilateral
Angle Divider Theorem for a Cyclic Quadrilateral
The quasi-circumcentre and quasi-incentre of a quadrilateral
Cyclic Hexagon Alternate Angles Sum Theorem
Circumscribed Hexagon Alternate Sides Theorem
Triangle Incentre-Circumcentre Collinearity
Visually Introducing & Classifying Quadrilaterals by Dragging (Grades 1-7)
An Inclusive, Hierarchical Classification of Quadrilaterals
Some External Links
Right kite (Wikipedia)
Isosceles tangential trapezoid (Wikipedia)
Bicentric quadrilateral (Wikipedia)
Bicentric Quadrilateral (Wolfram MathWorld)
********************************
Back to "Dynamic Geometry Sketches"
Back to "Student Explorations"
Michael de Villiers, created 18 April 2025 with WebSketchpad.