(1) Drawing the figure, I observe that ratios AE/AD and BC/BE are approximately equal, having a value of 1.6.

(2) From Pythagoras' Theorem I calculate MC (in the right-angled triangle MBC):

MC2 = BC2 + MB2 = 1 + 1/4 MC = Ö5 /2 .

So AE=(Ö5 + 1)/2 and BE=(Ö5 - 1)/2. The ratios are:

AE AD = Ö5 + 1 2

and

BC BE = 1 (Ö5 - 1)/2 = Ö5 + 1 2 .

So, AE/AD = BC/BE.

(3) From this equality of ratios, I find out that

BE = AD.BC AE = 1 f

But AE = AB + BE so

f = 1 + 1 f .

(4) Substituting f = 1 the left hand side of this expression is less than the right hand side. If we increase the value given to f the left hand side increases and the right hand side decreases continuously. Substituting f = 2 the left hand side is greater than the right hand side so the value of f which satisfies this equation must lie between 1 and 2.

The two solutions of the equation can be found at the intersection of the cyan curve (y=1 + 1/x) and magenta curve (y=x). Only the positive value is considered and it is approximately 1.618.

(5) Now, I solve the equation. It is equivalent to f² - f-1 = 0 so the solutions are

f1 = 1-Ö5 2

and

f1 = 1+Ö5 2 .

Only the second solution is valid because f > 0 .