### Golden Triangle

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.

### Pent

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

### Pentakite

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

### Why do this problem?

Provide training in conjecture, mathematical analysis and proof. This difficult problem requires students to realise that it is possible to use counting (a discrete process) to somehow categorise different paths. This gives the power to make general statements.

### Possible Approach?

First students need to calculate the end points of a few simple roads. Although there are rational and irrational endpoints, group discussion should lead to the conclusion that root 3 should be involved in the irrational endpoints in some way.

### Key Questions?

If you have a valid road, how can its endpoint change with the addition of a single tile? Two tiles?
Can you make any 'families' of roads which are similar, yet of different lengths? Can you create expressions for the lengths of these families of roads?