Circle properties and circle theorems

There are 50 NRICH Mathematical resources connected to Circle properties and circle theorems
circles in quadrilaterals
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Circles in quadrilaterals

Age
14 to 16
Challenge level
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Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Cyclic Quadrilaterals
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Cyclic quadrilaterals

Age
11 to 16
Challenge level
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Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Belt
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Belt

Age
16 to 18
Challenge level
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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.
Compare Areas
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Compare areas

Age
14 to 16
Challenge level
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Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?
Sitting Pretty
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Sitting pretty

Age
14 to 16
Challenge level
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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?
Baby Circle
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Baby circle

Age
16 to 18
Challenge level
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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?
Salinon
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Salinon

Age
14 to 16
Challenge level
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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?
Triangles in circles
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Triangles in circles

Age
11 to 14
Challenge level
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Can you find triangles on a 9-point circle? Can you work out their angles?

Subtended angles
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Subtended angles

Age
11 to 14
Challenge level
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What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?
Right angles
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Right angles

Age
11 to 14
Challenge level
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Can you make a right-angled triangle on this peg-board by joining up three points round the edge?