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.
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
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?
Which hexagons tessellate?
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Can you make sense of these three proofs of Pythagoras' Theorem?
Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.
Prove Pythagoras' Theorem using enlargements and scale factors.
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. . . .
A huge wheel is rolling past your window. What do you see?
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 see how this picture illustrates the formula for the sum of the first six cube numbers?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
Can you find the areas of the trapezia in this sequence?
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
What fractions can you divide the diagonal of a square into by simple folding?
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 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!
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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.
Can you use the diagram to prove the AM-GM inequality?
This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.
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?
Four jewellers share their stock. Can you work out the relative values of their gems?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
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.
Kyle and his teacher disagree about his test score - who is right?
Can you make sense of the three methods to work out the area of the kite in the square?
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?
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
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?
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.
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
Can you discover whether this is a fair game?
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
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. . . .
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .
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.
Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
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. . . .
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.