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

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

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Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

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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 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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Prove Pythagoras' Theorem using enlargements and scale factors.

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

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

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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.

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

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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. . . .

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

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.

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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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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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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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Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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

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

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

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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 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 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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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?

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

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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?

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

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

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

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

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. . . .

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Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.

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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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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?

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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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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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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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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.

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

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

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