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

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

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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. . . .

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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.

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

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

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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?

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

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

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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.

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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.

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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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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.

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

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

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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.

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

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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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Prove Pythagoras' Theorem using enlargements and scale factors.

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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?

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!

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

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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?

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

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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?

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

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Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

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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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I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

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

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When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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A huge wheel is rolling past your window. What do you see?

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

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The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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

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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. . . .

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Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

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Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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.

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There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

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Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.