Bicentric Quadrilateral Properties

A cyclic quadrilateral ABCD is circumscribed around a circle. (In general such quadrilaterals are called bicentric quadrilaterals).

What do you notice about the lines EG and FH joining the opposite tangential points?

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Bicentric Quadrilateral Properties

1) Click on the button in the sketch to construct lines EG and FH, and show the measure of the angle between them.

2) Drag the incentre I, circumcentre O or tangent point H to change the shape and to check your observation in 1) above.

3) Can you explain (prove) the result yourself?

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.)

4) Click on the second 'More Properties' button to show more properties. The property that I, O and the intersection of EG and FH are collinear, follows directly from this Lemma, but the property that the diagonals of ABCD are concurrent with the intersection of EG and FH, is most easily proved with projective geometry (e.g. see Aref, M.N. & Wernick, W. (1968). Problem & Solutions in Euclidean Geometry. Dover Publications: New York, p. 209.)

5) 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: Bicentric quadrilaterals have several other interesting properties, many of which can be found in books on advanced Euclidean geometry and in mathematical journals, or simply by searching the internet via Google.

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Michael de Villiers, updated 7 April 2011.