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. . . .
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.
Which hexagons tessellate?
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!
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
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.
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.
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?
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.
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.
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.
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
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?
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?
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Can you make sense of these three proofs of Pythagoras' Theorem?
An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?
What fractions can you divide the diagonal of a square into by simple folding?
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?
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?
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.
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.
Show that among the interior angles of a convex polygon there cannot be more than three acute angles.
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?
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?
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.
There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?
When is it impossible to make number sandwiches?
What is the largest number of intersection points that a triangle and a quadrilateral can have?
If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
Kyle and his teacher disagree about his test score - who is right?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
An article which gives an account of some properties of magic squares.
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Can you discover whether this is a fair game?
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?