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

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

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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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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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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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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?

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

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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 Pythagoras' Theorem using enlargements and scale factors.

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

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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.

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

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

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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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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 the three methods to work out the area of the kite in the square?

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

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

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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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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.

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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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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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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. . . .

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

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

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