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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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A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

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The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

Liethagoras, Pythagoras' cousin (!), was jealous of Pythagoras and came up with his own theorem. Read this article to find out why other mathematicians laughed at him.

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If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units.... What is the area of the triangle?

A description of some experiments in which you can make discoveries about triangles.

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Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. . . .

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A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

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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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The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and. . . .

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Two right-angled triangles are connected together as part of a structure. An object is dropped from the top of the green triangle where does it pass the base of the blue triangle?

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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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ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

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Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.

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Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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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?

Jennifer Piggott and Charlie Gilderdale describe a free interactive circular geoboard environment that can lead learners to pose mathematical questions.

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Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

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You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

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Take any point P inside an equilateral triangle. Draw PA, PB and PC from P perpendicular to the sides of the triangle where A, B and C are points on the sides. Prove that PA + PB + PC is a constant.

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If the yellow equilateral triangle is taken as the unit for area, what size is the hole ?

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Construct a line parallel to one side of a triangle so that the triangle is divided into two equal areas.

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Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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Using LOGO, can you construct elegant procedures that will draw this family of 'floor coverings'?

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Start with a triangle. Can you cut it up to make a rectangle?

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Find the area of the shaded region created by the two overlapping triangles in terms of a and b?

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Determine the total shaded area of the 'kissing triangles'.

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From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.

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A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...

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Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

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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?

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Make an equilateral triangle by folding paper and use it to make patterns of your own.

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Can you make a regular hexagon from yellow triangles the same size as a regular hexagon made from green triangles ?

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A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?

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Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.

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Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.

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Can you work out the fraction of the original triangle that is covered by the inner triangle?

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Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?

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Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?