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This question involves the sides of a right-angled triangle, the Golden Ratio, and the arithmetic, geometric and harmonic means of two numbers. Take any two numbers a and b, where 0 < b < a .
The arithmetic mean (AM) is (a+b)/2 ;
the geometric mean (GM) is \sqrt{ab} ;
the harmonic mean (HM) is {1 \over {{1 \over 2}\left( {1 \over a} + {1\over b } \right)}};
Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.
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.