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:

The second part of this question introduces an expression involving
an infinite string of nested square roots but, as forbidding as it
may seem, a little ingenuity is all that is needed to evaluate it.