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Bicentric Quadrilateral: Definition & Properties

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will find out what a bicentric quadrilateral is and its properties and characteristics. We will also look at an example of using properties of bicentric quadrilaterals to find certain aspects of the quadrilateral.

Bicentric Quadrilateral

Suppose that designers at Tall Oak Toys are trying to create a new type of kite that is the shape of a typical kite (in mathematics, this is called a right kite), but it has a circle inside the kite that touches each of the sides of the kite at exactly one point, and a circle on the outside of the kite that passes through all of the vertices of the kite.


Kite Toy
diagram of kite with one circle inside and one circle surrounding


In mathematics, we have special names for each of the parts of this kite. First of all, the kite is a quadrilateral, because it is a four-sided polygon. Second, the circles have special names as well.

  • Incircle: The incircle of a polygon is a circle that can be drawn on the inside of the polygon that touches each of the sides of the polygon exactly once.
  • Circumcircle: The circumcircle of a polygon is a circle that is drawn around the polygon that passes through all of the polygons vertices.

Now, putting this all together, we call any quadrilateral that has both an incircle and a circumcircle a bicentric quadrilateral. Therefore, this kite that the designers are creating is a bicentric quadrilateral. That's a pretty fancy name for a pretty simple concept!

Now, suppose the designers are trying to figure out the lengths that the radii of both of the circles should be in order to build the kites properly so they will be able to be flown.

Once again, these radii have special names in mathematics!

  • Inradius: The inradius of a quadrilateral is the radius of the incircle of the quadrilateral.
  • Circumradius: The circumradius of a quadrilateral is the radius of the circumcircle of the quadrilateral.
  • Offset: The line segment connecting the center of the incircle and the center of the circumcircle.


Bicentric Quadrilateral
kite diagram with labeled parts


Properties of Bicentric Quadrilaterals

We know the lengths of the sides of the kite, but how in the heck are we going to find the lengths of those radii? Thankfully, bicentric quadrilaterals have a couple of really neat properties that make finding these lengths into a simple matter of using a formula.

Let's start with the inradius of a bicentric quadrilateral. To find the inradius of a bicentric quadrilateral with side lengths a, b, c, and d, we use the following formula:

r = √(abcd) / ((1/2)(a+b+c + d))

Now, consider the circumradius of a bicentric quadrilateral. To find the circumradius of a bicentric quadrilateral we use the following formula:

R= (1/4)(√(((ac + bd)(ad+bc)(ab + cd)) / (abcd))

Ah! Formulas can make things so much easier! Let's use these to find the lengths of the radii of the kite, so the designers can get these things into production!

Example

The kite that the designers are creating has side lengths a = 18 in, b = 27 in, c = 27 in, and d = 18 in. Therefore, we have all we need to find the inradius and circumradius of the kite. First, we'll find the inradius. We plug our side lengths into the formula and simplify.


example problem


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