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What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?

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A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

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Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?

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

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ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.

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For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

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A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

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Describe how to construct three circles which have areas in the ratio 1:2:3.

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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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Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

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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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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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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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This set of resources for teachers offers interactive environments which support work on properties of angles in circles at Key Stage 4.

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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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Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

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A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?

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A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.

This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.

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Can you find a relationship between the area of the crescents and the area of the triangle?

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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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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?

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

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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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What is the same and what is different about these circle questions? What connections can you make?

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The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

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Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

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Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.

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Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

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M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

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A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

This gives a short summary of the properties and theorems of cyclic quadrilaterals and links to some practical examples to be found elsewhere on the site.

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Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

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A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

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

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Use trigonometry to determine whether solar eclipses on earth can be perfect.