Circle properties and circle theorems

problem
Set square

Age

16 to 18

Challenge level

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?
problem
Kissing

Age

16 to 18

Challenge level

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?
problem
Fixing it

Age

16 to 18

Challenge level

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?
problem
Baby circle

Age

16 to 18

Challenge level

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?
problem
Some(?) of the parts

Age

14 to 16

Challenge level

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
page
Circle theorems

This set of resources for teachers offers interactive environments which support work on properties of angles in circles at Key Stage 4.
article
Interacting with the geometry of the circle

Jennifer Piggott and Charlie Gilderdale describe a free interactive circular geoboard environment that can lead learners to pose mathematical questions.
article
Incircles explained

This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.
article
The cyclic quadrilateral

This gives a short summary of the properties and theorems of cyclic quadrilaterals and links to some practical examples to be found elsewhere on the site.
interactivity
Virtual geoboard

A virtual geoboard that allows you to create shapes by stretching rubber bands between pegs on the board. Allows a variable number of pegs and variable grid geometry and includes a point labeller.