### Floored

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

### Pie Cuts

Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).

### Getting an Angle

How can you make an angle of 60 degrees by folding a sheet of paper twice?

# Terminology

##### Stage: 3 Challenge Level:

Correct solutions from Mary of Birchwood High School and Andrei of School 205 Bucharest have contributed to the edited solution below.

As triangle DEF is equilateral, all its angles are $60$ .

Angle AEF $= 180 - 60 - c = 120 - c$

Similarly

Angle BFD $= 120 - b$
Angle EDC $= 120 - a$

From triangle FAE, I calculate angle A:

As triangle ABC is an isosceles Angle ABC $=$ Angle ACB $= x$

Therefore $180 - (120 - b + a ) = 180 - (120 - a + c )$
Therefore $60 + b - a = 60 + a - c$
Therefore $2a = b + c$
Therefore $a = (b + c)/2$

Part Two

If $a=b=c$

Angle ABC $= 180 - (120 - b + a ) = 180 - (120 - a + a ) = 60$
Angle ACB $= 180 - (120 - a + c ) = 180 - (120 - a + a ) = 60$
Therefore angle BAC is $60$

Therefore triangle ABC is equilateral.