Golden ratio

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

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.

Explore the geometry of these dart and kite shapes!

You add 1 to the golden ratio to get its square. How do you find higher powers?

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.

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.

Given a regular pentagon, can you find the distance between two non-adjacent vertices?

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!