### Overturning Fracsum

Solve the system of equations to find the values of x, y and z: xy/(x+y)=1/2, yz/(y+z)=1/3, zx/(z+x)=1/7

### Bang's Theorem

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

### Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

# Polycircles

##### Stage: 4 Challenge Level:

Try a numerical example, say the 3 points give a triangle of sides 12, 15 and 13. Find the radii of the circles.

Now use the same method if the sides of the triangle are $a$, $b$ and $c$?

Generalise still more: What about 4 circles? 5 circles? $n$ circles?