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.

Golden Eggs

Age 16 to 18 Challenge Level:

(1) You will need to know, or find out, the formula for the area of
an ellipse. All you have to do then is solve a quadratic equation.

(2) What happens if you square this strange
expression?