Circle properties and circle theorems

There are 50 NRICH Mathematical resources connected to Circle properties and circle theorems
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?
Lens Angle
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Lens angle

Age
14 to 16
Challenge level
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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.
Partly Circles
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Partly circles

Age
14 to 16
Challenge level
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What is the same and what is different about these circle questions? What connections can you make?
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?