Challenge Level

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. . . .

Challenge Level

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?

Challenge Level

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

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Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

Challenge Level

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

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

Challenge Level

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

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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?

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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?

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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?

Challenge Level

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?

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

Challenge Level

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.

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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.

Challenge Level

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.

Challenge Level

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.

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

Challenge Level

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

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!

Challenge Level

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.

Challenge Level

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

Challenge Level

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

Challenge Level

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

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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.

Challenge Level

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

Challenge Level

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

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

Challenge Level

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?

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

Here are some examples of 'cons', and see if you can figure out where the trick is.

Challenge Level

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

Challenge Level

Prove Pythagoras' Theorem using enlargements and scale factors.

Challenge Level

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.

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

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

Challenge Level

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?

Challenge Level

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

Challenge Level

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. . . .

Challenge Level

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

Challenge Level

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?

Challenge Level

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. . . .

Challenge Level

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

Challenge Level

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. . . .

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

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

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Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

Challenge Level

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

Challenge Level

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?