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Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

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Circumspection

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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Quads

The circumcentres of four triangles are joined to form a quadrilateral. What do you notice about this quadrilateral as the dynamic image changes? Can you prove your conjecture?

Compare Areas

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem ?

This problem brings together ideas of areas of circles and squares, the use of Pythagoras theorem and the property of tangents to a circle from an exernal point.

Possible approach

You might start with the middle diagram which is the easiest. It brings in the ratio of the sides of an isosceles right angles triangle which is again used in the other two parts.

Key questions

If you know the side length of an isosceles right angled triange how do you find the hypotenuse?

Which lengths are equal in the digram?

Which angles are equal?

Can you use the symmetry of the diagram?

Possible extension

The problem Circle-in also uses one of the circle theorems (the tangent is perpendicular to the radius at the point of contact).