### Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

##### Stage: 4 Challenge Level:

First, try the problem Circles in Quadrilaterals to familiarise yourself with the properties of tangential quadrilaterals.

A bicentric quadrilateral is both tangential and cyclic. In other words, it is possible to draw a circle inside it which touches all four sides, and also to draw another circle around it which passes through all four vertices. (The two circles do not necessarily have the same centre!)

Here is a picture of a bicentric quadrilateral.
Think about special quadrilaterals, such as squares, trapezia, and parallelograms.
• Which types of quadrilateral are always bicentric?
• Which types of quadrilateral are never bicentric?
• For the quadrilaterals which are sometimes bicentric, can you explain the conditions necessary for them to be bicentric?

There is a formula for finding the area $A$ of a bicentric quadrilateral:
$$A = \sqrt{abcd}$$

where $a,b,c$ and $d$ are the lengths of the four sides.

Verify that this formula gives the correct area for the examples of bicentric quadrilaterals you have found.