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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

Many got the correct answer that the circle which is embedded inside the isosceles triangle is the largest of the three shapes.

Some students measured the diagrams given which luckily had been drawn accurately to scale, while others observed the symmetry of the isosceles triangle and the consequences of joining B to D the mid point of AC. To find the radius of the circle it helps to draw perpendiculars from the centre to the sides of the triangle.

Taking the lengths of the short sides of the triangle as $2$ units:

the radius of the circle is found to be $2-\sqrt{2}$ and the area to be $\pi (6 - 4\sqrt{2})$.

the side length of square one to be $1$ and area $1$;

and the side length of square two to be $\frac{2\sqrt{2}}{3}$ and area $\frac{8}{9}$.

Many students evaluated the ratio of the areas as $1.078 : 1: 0.889$ (to $3$ dec. places)

Ben and Adrianfrom the Simon Langton Grammar School for Boys used trig ratios, as did Hannah of Maidstone Girls Grammar School, to arrive at the same conclusions.