It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

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?

This is the second article on right-angled triangles whose edge lengths are whole numbers.

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.

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?

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 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!

Can you make sense of these three proofs of Pythagoras' Theorem?

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

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.

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

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.

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

Can you find the areas of the trapezia in this sequence?

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. . . .

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?

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.

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?

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

Can you make sense of the three methods to work out the area of the kite in the square?

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

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 jewellers share their stock. Can you work out the relative values of their gems?

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

Prove Pythagoras' Theorem using enlargements and scale factors.

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.

Do you have enough information to work out the area of the shaded quadrilateral?

This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

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.

When is it impossible to make number sandwiches?

Keep constructing triangles in the incircle of the previous triangle. What happens?

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

What is the largest number of intersection points that a triangle and a quadrilateral can have?

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?

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Kyle and his teacher disagree about his test score - who is right?

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

Can you explain why a sequence of operations always gives you perfect squares?

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.