Prove this geometric mean?

I've revised the 2nd paragraph to work with the lengths in the second image, per your request:

First I observe that for positive x and N, x+N/x is minimized when x=√N.

This is easy to see if we rewrite x+N/x = (√x - √N/√x)² + 2 and since the square term is not negative, the expression is minimzed when the square term is 0, which happens only for x=√N.

Now I note that Φ = arctan(b/c) - arctan (a/c).

Remembering that tan(x-y) = (tan(x) - tan(y)) / (1 + tan(x) tan(y)) gives

tan (Φ) = (b/c - a/c) / (1 + ab/c²) = (a+b) / (c + ab/c),

and since the tangent is an increasing function, it is maximized

(and hence Φ is maximized) when the denominator in (a+b) / (c + ab/c)

is minimized, which, by the prior paragraph, happens when c² = ab.