### Some(?) of the Parts

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

### Triangle Incircle Iteration

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

### Medallions

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

# Compare Areas

##### Age 14 to 16Challenge Level

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