The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Which hexagons tessellate?
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?
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.
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
Prove Pythagoras' Theorem using enlargements and scale factors.
Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.
What fractions can you divide the diagonal of a square into by simple folding?
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
Can you find the areas of the trapezia in this sequence?
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
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.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.
Can you make sense of these three proofs of Pythagoras' Theorem?
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
Can you make sense of the three methods to work out the area of the kite in the square?
Show that among the interior angles of a convex polygon there cannot be more than three acute angles.
Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.
It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
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?
Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Whenever two chameleons of different colours meet they change colour to the third colour. Describe the shortest sequence of meetings in which all the chameleons change to green if you start with 12. . . .
I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
Four jewellers share their stock. Can you work out the relative values of their gems?
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.
What is the largest number of intersection points that a triangle and a quadrilateral can have?
When is it impossible to make number sandwiches?
Kyle and his teacher disagree about his test score - who is right?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...