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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

Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Circumspection

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

Bicentric Quadrilaterals

Age 14 to 16 Challenge Level:

Why do this problem

This problem provides an opportunity to consider properties of quadrilaterals and circle theorems while investigating the unfamiliar and intriguing idea of bicentric quadrilaterals.

Possible approach

First, try the problem Circles in Quadrilaterals. For learners who are unfamiliar with the properties of cyclic quadrilaterals, the problems Triangles in Circles and Pegboard Quads can be used to investigate.

Explain that a bicentric quadrilateral is both cyclic and tangential, so a circle can be drawn inside it just touching each side, and another circle (not necessarily with the same centre) can be drawn around it just touching each vertex. Ask learners to sketch examples of bicentric quadrilaterals, and encourage them to share ideas in pairs about how to identify which quadrilaterals are bicentric.

Bring the class together for discussion of their ideas. One hint for finding bicentric quadrilaterals is to start with a tangential one and make it also cyclic, or vice versa. Experimenting with a dynamic geometry package such as Geogebra can give some insight into the properties of these quadrilaterals.

Then give pairs or small groups time to construct some examples of bicentric quadrilaterals and calculate their areas. The areas can then be used to verify the area formula given for these cases.

Key questions

What properties must a quadrilateral have to be tangential?
What properties must a quadrilateral have to be cyclic?
Which quadrilaterals can be both tangential and cyclic at the same time?

Possible extension

Show that the area formula given will hold for specific types of quadrilateral such as squares and kites, given the constraint that they are bicentric. Proving the formula generally is extremely challenging!

Possible support

Try Triangles in Circles as a basis for investigating circle theorems.