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. . . .

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?

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.

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

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!

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.

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

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?

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

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

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.

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.

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?

What fractions can you divide the diagonal of a square into by simple folding?

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.

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

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?

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

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.

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

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

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

Prove Pythagoras' Theorem using enlargements and scale factors.

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

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

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.

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

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

When is it impossible to make number sandwiches?

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

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?

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?

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?

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.

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

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

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

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

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

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

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

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. . . .

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry