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.
Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'. Show that the sum of the areas of the two crescents equals the area of triangle ABC.
The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
This set of resources for teachers offers interactive environments
which support work on properties of angles in circles at Key Stage
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
You are given a circle with centre O. Describe how to construct with a straight edge and a pair of compasses, two other circles centre O so that the three circles have areas in the ratio 1:2:3.
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.
What can you say about the angles on opposite vertices of any
cyclic quadrilateral? Working on the building blocks will give you
insights that may help you to explain what is special about them.
In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .
What is the same and what is different about these circle
questions? What connections can you make?
How many different triangles can you make which consist of the
centre point and two of the points on the edge? Can you work out
each of their angles?
What is the relationship between the angle at the centre and the
angles at the circumference, for angles which stand on the same
arc? Can you prove it?
A small circle in a square in a big circle in a trapezium. Using
the measurements and clue given, find the area of the trapezium.
Two intersecting circles have a common chord AB. The point C moves
on the circumference of the circle C1. The straight lines CA and CB
meet the circle C2 at E and F respectively. As the point C. . . .
A circle has centre O and angle POR = angle QOR. Construct tangents
at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q
lie inside, or on, or outside this circle?
A picture is made by joining five small quadrilaterals together to
make a large quadrilateral. Is it possible to draw a similar
picture if all the small quadrilaterals are cyclic?
The circumcentres of four triangles are joined to form a
quadrilateral. What do you notice about this quadrilateral as the
dynamic image changes? Can you prove your conjecture?
Two circles intersect at A and B. Points C and D move round one
circle. CA and DB cut the other circle at E and F. What do you
notice about the line segments CD and EF?
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?
In the diagram the point P can move to different places around the
dotted circle. Each position P takes will fix a corresponding
position for P'. As P moves around on that circle what will P' do?
In the diagram the point P' can move to different places along the
dotted line. Each position P' takes will fix a corresponding
position for P. If P' moves along a straight line what does P do ?
Can you make a right-angled triangle on this peg-board by joining
up three points round the edge?
Jennifer Piggott and Charlie Gilderdale describe a free interactive
circular geoboard environment that can lead learners to pose
ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
An equilateral triangle is sitting on top of a square.
What is the radius of the circle that circumscribes this shape?
Four rods of equal length are hinged at their endpoints to form a
rhombus. The diagonals meet at X. One edge is fixed, the opposite
edge is allowed to move in the plane. Describe the locus of. . . .
Find the missing angle between the two secants to the circle when
the two angles at the centre subtended by the arcs created by the
intersections of the secants and the circle are 50 and 120 degrees.
Points D, E and F are on the the sides of triangle ABC.
Circumcircles are drawn to the triangles ADE, BEF and CFD
respectively. What do you notice about these three circumcircles?